Posts by Radu Precup

Abstract

In this paper we discuss the localization and the multiplicity of solutions for the stationary Stokes system with variable viscosity and a reaction force term. The results obtained apply to systems with strongly oscillating periodic viscosity and the corresponding homogenized systems.

Authors

Renata Bunoiu
University of Lorraine, Metz, France

Radu Precup
Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babes-Bolyai, University, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania

Keywords

Stokes system, localization and multiplicity of solutions, fixed point method, homogenization

Paper coordinates

R. Bunoiu, R. Precup, Localization and multiplicity for stationary Stokes systems with variable viscosity, Communications in Mathematical Analysis and Applications, (2025), 1-28, https://dx.doi.org/10.2139/ssrn.4887415

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Communications in Mathematical Analysis and Applications
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Localization and multiplicity for stationary Stokes systems with variable viscosity

Renata BunoiusuperscriptBunoiu\text{Bunoiu}^{*}Bunoiu start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT
University of Lorraine, CNRS, IECL, F-57000, Metz, France
renata.bunoiu@univ-lorraine.fr
   Radu Precup
Institute of Advanced Studies in Science and Technology STAR-UBB,
Babeş-Bolyai University &
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy,
Cluj-Napoca, Romania
r.precup@math.ubbcluj.ro
Abstract

In this paper we discuss the localization and the multiplicity of solutions for the stationary Stokes system with variable viscosity and a reaction force term. The results obtained apply to systems with strongly oscillating periodic viscosity and the corresponding homogenized systems.

1112020 Mathematics Subject Classification : 35Q30, 47J05 Key words: Stokes system, localization and multiplicity of solutions, fixed point method, homogenization * corresponding author

1 Introduction

In fluid mechanics, Stokes system describes the linear flow of a viscous fluid. The case of a constant viscosity was extensively studied in the literature (see for instance [14], [10], and [27]). The aim of this paper is to study the localization, multiplicity and homogenization for stationary Stokes systems with variable viscosity in bounded domains. The viscosity is a characteristic of the nature of the fluid and in our case it is supposed to depend on the position. This describes the variable properties of the fluid, as for instance in the case of complex multiphase flow. As applications of such flow one can mention a wide range of engineering problems such as power generation, water treatment, water desalination, refrigeration and air conditioning, and carbon capture and sequestration. We may refer the reader to [20], [8], [11] for the Stokes flow with variable viscosity in thin domains, and to [1], [2] for homogenization results.

We start our study with the Stokes system (2.1) in pseudo-stress-velocity formulation (see for instance [6], [7], and [16]), modeling the steady-state motion under an external force f𝑓fitalic_f, of an incompressible fluid with variable viscosity μ𝜇\muitalic_μ, where u𝑢uitalic_u represents the velocity of the fluid and p𝑝pitalic_p is the pressure. Our first objective is to study the localization and the multiplicity of the solution for a system similar to (2.1), if the right-hand side in the first equation depends in a nonlinear manner on the velocity (see (2.11)). From the physical point of view, this corresponds to a reaction external force. A first practical application of our study is to find a velocity dependent reaction force which guarantees that each component of the velocity stays bounded between two a priori given bounds. A second application is to find bounds for the velocity when the reaction external force is given.

In general, two essential tools for proving the localization and multiplicity are the maximum principle and Moser-Harnack type inequalities (see [23]). In order to compensate the loss of these tools in the case of Stokes system, the idea is to associate to system (2.1) (and thus to system (2.11)) a diffusion problem. To do so, we notice that in (2.1) the term p𝑝\nabla p∇ italic_p can be seen as a correction of the external force f𝑓fitalic_f, in order to keep the incompressibility of the fluid. Through the system (2.6) we associate to p𝑝\nabla p∇ italic_p the function q𝑞qitalic_q, which can be seen as a correction of the velocity u𝑢uitalic_u and then we define w=u+q𝑤𝑢𝑞w=u+qitalic_w = italic_u + italic_q, which solves the diffusion problem (2.7) and so w𝑤witalic_w can be interpreted as the recovered velocity for problem (2.1), under the external force f𝑓fitalic_f and a constant pressure. Consequently, following this idea, we can match a Stokes system with variable viscosity and a corresponding diffusion problem with diffusion coefficients varying with space (see for instance [5]). Once we do this, we prove localization and multiplicity results for the diffusion problem, which will give localization and multiplicity results for the recovered velocity of the Stokes system.

Except for the case of the Navier-Stokes system, where the velocity-dependent convective term can be seen as a reaction force, and for the case of the Coriolis force (see, e.g., [9], [17]), up to our knowledge, there are few mathematical studies in the literature on Stokes systems with a more general reaction force. In this paper we first consider nonlinear reaction forces of the type h(x,u(x)+q(x))𝑥𝑢𝑥𝑞𝑥\ h\left(x,u\left(x\right)+q\left(x\right)\right)italic_h ( italic_x , italic_u ( italic_x ) + italic_q ( italic_x ) ) (see (2.11)). Thus the external force hhitalic_h is sensitive to the recovered velocity w=u+q𝑤𝑢𝑞\ w=u+q\ italic_w = italic_u + italic_q incorporating both velocity u𝑢uitalic_u and pressure p𝑝pitalic_p (the last one by means of q𝑞qitalic_q, which is a correction of the velocity). The main result concerning this first system in given in Theorem 4. Then, we consider the special case of planar Stokes systems with a constant pressure, namely such that p=0.𝑝0\nabla p=0.∇ italic_p = 0 . We start by giving a sufficient condition on the right-hand side, which guarantees that the pressure of the planar Stokes system is constant. Then we are able to discuss the case of velocity-dependent reaction forces, to localize the velocity u𝑢uitalic_u itself, and to obtain multiple solutions with distinct velocity components. The main result concerning this second case in given in Theorem 11. Following the ideas in our previous work [4], we end this paper by giving localization and multiplicity results for the two previous cases, under the hypothesis that the viscosity is highly heterogeneous, being an ε𝜀\varepsilonitalic_ε-periodic function on each spatial direction, with ε𝜀\varepsilonitalic_ε a small positive parameter that we will let tend to zero. From the physical point of view, this situation modelizes for instance a particular mixture of fluids. We refer to [1] for the homogenization of the Stokes system in pseudo-stress-velocity formulation with ε𝜀\varepsilonitalic_ε-periodic viscosity. For homogenization problems with nonlinear reaction terms we refer for instance to [19] and [12]. Localization and multiplicity results for a nonlinear right-hand side in the homogenization context are given in Theorem 15 and Theorem 17.

The paper is organized as follows. In Section 2 we state the problem and we give some preliminary results. Section 3 deals with the localization and multiplicity results for the two cases under study. In Section 4 we give analogous results in the periodic homogenization context.

2 Statement of the problem and preliminary results

We consider the mathematical model for the steady-state motion of an incompressible fluid with variable viscosity given by the pseudo-stress-velocity formulation Stokes system (2.1) below

{div(μ(x)u)+p=f(x)in Ωdivu=0in Ωu=0on Ω.casesdiv𝜇𝑥𝑢𝑝𝑓𝑥in Ωdiv𝑢0in Ω𝑢0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu\left(x\right)\nabla u% \right)+\nabla p=f\left(x\right)&\text{in }\Omega\\ \operatorname{div}\,u=0&\text{in }\Omega\\ u=0&\text{on }\partial\Omega.\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ ( italic_x ) ∇ italic_u ) + ∇ italic_p = italic_f ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u = 0 end_CELL start_CELL on ∂ roman_Ω . end_CELL end_ROW end_ARRAY (2.1)

Here ΩnΩsuperscript𝑛\Omega\subset\mathbb{R}^{n}\ roman_Ω ⊂ blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT (n2𝑛2n\geq 2italic_n ≥ 2)  is a C2superscript𝐶2C^{2}italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bounded open set, μC1(Ω¯)𝜇superscript𝐶1¯Ω\mu\in C^{1}\left(\overline{\Omega}\right)italic_μ ∈ italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( over¯ start_ARG roman_Ω end_ARG ) with 0<μ¯μ(x)μ¯0¯𝜇𝜇𝑥¯𝜇0<\underline{\mu}\leq\mu\left(x\right)\leq\overline{\mu}0 < under¯ start_ARG italic_μ end_ARG ≤ italic_μ ( italic_x ) ≤ over¯ start_ARG italic_μ end_ARG and f=(f1,..,fn)L2(Ω;n)f=\left(f_{1},..,f_{n}\right)\in L^{2}\left(\Omega;\mathbb{R}^{n}\right)italic_f = ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , . . , italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ). Physically, there are relevant the cases n=2𝑛2n=2italic_n = 2 and n=3,𝑛3n=3,italic_n = 3 , where μ(x)𝜇𝑥\mu\left(x\right)italic_μ ( italic_x ) stands for the variable viscosity of the fluid, f(x)𝑓𝑥f\left(x\right)italic_f ( italic_x ) for the external force, the unknown functions u𝑢uitalic_u and p𝑝pitalic_p are the velocity and pressure, respectively, while the condition divu=i=1nxiui=0div𝑢superscriptsubscript𝑖1𝑛subscriptsubscript𝑥𝑖subscript𝑢𝑖0\operatorname{div}\,u=\sum_{i=1}^{n}\partial_{x_{i}}u_{i}=0roman_div italic_u = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 means that the fluid is incompressible. On the boundary of the domain ΩΩ\Omegaroman_Ω we consider no-slip condition.

It is known (see, e.g., [10], [14], [18], [25], [27]) that a weak solution exists and is unique, that is a pair (u,p)𝑢𝑝\left(u,p\right)( italic_u , italic_p ) with uH01(Ω;n),𝑢superscriptsubscript𝐻01Ωsuperscript𝑛u\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right),italic_u ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) , divu=0,div𝑢0\operatorname{div}\,u=0,roman_div italic_u = 0 , pL2(Ω)𝑝superscript𝐿2Ωp\in L^{2}\left(\Omega\right)italic_p ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ), p𝑝pitalic_p being unique up to an additive constant, and

((u,v))H01+(p,divv)L2=(f,v)L2for all vH01(Ω;n),formulae-sequencesubscript𝑢𝑣superscriptsubscript𝐻01subscript𝑝div𝑣superscript𝐿2subscript𝑓𝑣superscript𝐿2for all 𝑣superscriptsubscript𝐻01Ωsuperscript𝑛\left(\left(u,v\right)\right)_{H_{0}^{1}}+\left(p,\operatorname{div}\,v\right)% _{L^{2}}=\left(f,v\right)_{L^{2}}\ \ \ \text{for all }v\in H_{0}^{1}\left(% \Omega;\mathbb{R}^{n}\right),( ( italic_u , italic_v ) ) start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT + ( italic_p , roman_div italic_v ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ( italic_f , italic_v ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT for all italic_v ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) , (2.2)

where ((u,v))H01=Ωμ(x)uvdxsubscript𝑢𝑣superscriptsubscript𝐻01subscriptΩ𝜇𝑥𝑢𝑣𝑑𝑥~{}\left(\left(u,v\right)\right)_{H_{0}^{1}}=\int_{\Omega}\mu\left(x\right)% \nabla u\cdot\nabla v\,dx( ( italic_u , italic_v ) ) start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT italic_μ ( italic_x ) ∇ italic_u ⋅ ∇ italic_v italic_d italic_x. By uv𝑢𝑣\ \nabla u\cdot\nabla v∇ italic_u ⋅ ∇ italic_v we mean i=1nuivisuperscriptsubscript𝑖1𝑛subscript𝑢𝑖subscript𝑣𝑖\ \sum\limits_{i=1}^{n}\nabla u_{i}\cdot\nabla v_{i}∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∇ italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⋅ ∇ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and (f,v)L2=i=1n(fi,vi)L2.subscript𝑓𝑣superscript𝐿2superscriptsubscript𝑖1𝑛subscriptsubscript𝑓𝑖subscript𝑣𝑖superscript𝐿2\ \left(f,v\right)_{L^{2}}=\sum\limits_{i=1}^{n}\left(f_{i},v_{i}\right)_{L^{2% }}.~{}( italic_f , italic_v ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT .Also u𝑢uitalic_u is the unique function in V={vH01(Ω;n):divv=0}𝑉conditional-set𝑣superscriptsubscript𝐻01Ωsuperscript𝑛div𝑣0V=\left\{v\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right):\ \operatorname{div}% \,v=0\right\}italic_V = { italic_v ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) : roman_div italic_v = 0 } satisfying

((u,v))H01(f,v)L2=0for all vV.formulae-sequencesubscript𝑢𝑣superscriptsubscript𝐻01subscript𝑓𝑣superscript𝐿20for all 𝑣𝑉\left(\left(u,v\right)\right)_{H_{0}^{1}}-\left(f,v\right)_{L^{2}}=0\ \ \ % \text{for all \ }v\in V.( ( italic_u , italic_v ) ) start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - ( italic_f , italic_v ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0 for all italic_v ∈ italic_V . (2.3)

Indeed, for any pL2(Ω)𝑝superscript𝐿2Ωp\in L^{2}\left(\Omega\right)italic_p ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) one has p𝑝\nabla p∇ italic_p H1(Ω;n)absentsuperscript𝐻1Ωsuperscript𝑛\in H^{-1}\left(\Omega;\mathbb{R}^{n}\right)∈ italic_H start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) and (p,v)=(p,divv)L2=0𝑝𝑣subscript𝑝div𝑣superscript𝐿20\left(\nabla p,v\right)=\left(p,\operatorname{div}\,v\right)_{L^{2}}=0( ∇ italic_p , italic_v ) = ( italic_p , roman_div italic_v ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0 for all vV.𝑣𝑉v\in V.italic_v ∈ italic_V . Thus we may define S𝑆Sitalic_S, the velocity operator given by

S:L2(Ω;n)V,Sf:=uf,:𝑆formulae-sequencesuperscript𝐿2Ωsuperscript𝑛𝑉assign𝑆𝑓subscript𝑢𝑓S:L^{2}\left(\Omega;\mathbb{R}^{n}\right)\rightarrow V,\ \ \ \ \ Sf:=u_{f},italic_S : italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) → italic_V , italic_S italic_f := italic_u start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT , (2.4)

where ufsubscript𝑢𝑓u_{f}italic_u start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT is the unique solution in V𝑉Vitalic_V solving problem (2.3).

In addition, according for instance to [14, Theorem 5.3], we have that if J(v,q)𝐽𝑣𝑞J\left(v,q\right)italic_J ( italic_v , italic_q ) is defined by

J(v,q)=Ω(12μ(x)|v|2fvqdivv)𝑑x(vH01(Ω;n),qL2(Ω)),𝐽𝑣𝑞subscriptΩ12𝜇𝑥superscript𝑣2𝑓𝑣𝑞div𝑣differential-d𝑥formulae-sequence𝑣superscriptsubscript𝐻01Ωsuperscript𝑛𝑞superscript𝐿2ΩJ\left(v,q\right)=\int_{\Omega}\left(\frac{1}{2}\mu\left(x\right)\left|\nabla v% \right|^{2}-f\cdot v-q\operatorname{div}\,v\right)dx\ \ \ \left(v\in H_{0}^{1}% \left(\Omega;\mathbb{R}^{n}\right),\ q\in L^{2}\left(\Omega\right)\right),italic_J ( italic_v , italic_q ) = ∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_μ ( italic_x ) | ∇ italic_v | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_f ⋅ italic_v - italic_q roman_div italic_v ) italic_d italic_x ( italic_v ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) , italic_q ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) ) ,

then

J(u,p)=infvH01(Ω;n)supqL2(Ω)J(v,q)𝐽𝑢𝑝subscriptinfimum𝑣superscriptsubscript𝐻01Ωsuperscript𝑛subscriptsupremum𝑞superscript𝐿2Ω𝐽𝑣𝑞J\left(u,p\right)=\inf_{v\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right)}\sup_% {q\in L^{2}\left(\Omega\right)}J\left(v,q\right)italic_J ( italic_u , italic_p ) = roman_inf start_POSTSUBSCRIPT italic_v ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT roman_sup start_POSTSUBSCRIPT italic_q ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT italic_J ( italic_v , italic_q )

and u𝑢uitalic_u minimizes in V𝑉Vitalic_V the functional

JV(v)=Ω(12μ(x)|v|2fv)𝑑x(vV).subscript𝐽𝑉𝑣subscriptΩ12𝜇𝑥superscript𝑣2𝑓𝑣differential-d𝑥𝑣𝑉J_{V}\left(v\right)=\int_{\Omega}\left(\frac{1}{2}\mu\left(x\right)\left|% \nabla v\right|^{2}-f\cdot v\right)dx\ \ \ \left(v\in V\right).italic_J start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT ( italic_v ) = ∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_μ ( italic_x ) | ∇ italic_v | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_f ⋅ italic_v ) italic_d italic_x ( italic_v ∈ italic_V ) .

Denoting Lv:=div(μ(x)v),assign𝐿𝑣div𝜇𝑥𝑣\ Lv:=-\operatorname{div}\left(\mu\left(x\right)\nabla v\right),italic_L italic_v := - roman_div ( italic_μ ( italic_x ) ∇ italic_v ) , we define the operator L1:H1(Ω;n)H01(Ω;n):superscript𝐿1superscript𝐻1Ωsuperscript𝑛superscriptsubscript𝐻01Ωsuperscript𝑛\ L^{-1}:H^{-1}\left(\Omega;\mathbb{R}^{n}\right)\rightarrow H_{0}^{1}\left(% \Omega;\mathbb{R}^{n}\right)italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) → italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) where L1f=vf,superscript𝐿1𝑓subscript𝑣𝑓\ L^{-1}f=v_{f},\ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f = italic_v start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT , with vfH01(Ω;n)subscript𝑣𝑓superscriptsubscript𝐻01Ωsuperscript𝑛\ v_{f}\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right)italic_v start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) solving the Dirichlet problem

{div(μ(x)vf)=f(x)in Ωv=0on Ω.casesdiv𝜇𝑥subscript𝑣𝑓𝑓𝑥in Ω𝑣0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu\left(x\right)\nabla v_{% f}\right)=f\left(x\right)&\text{in }\Omega\\ v=0&\text{on }\partial\Omega.\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ ( italic_x ) ∇ italic_v start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT ) = italic_f ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_v = 0 end_CELL start_CELL on ∂ roman_Ω . end_CELL end_ROW end_ARRAY (2.5)

Letting q:=L1passign𝑞superscript𝐿1𝑝\ q:=L^{-1}\nabla pitalic_q := italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∇ italic_p for p𝑝\nabla p∇ italic_p as in (2.1), one has

{div(μ(x)q)=pin Ωq=0on Ω.casesdiv𝜇𝑥𝑞𝑝in Ω𝑞0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu\left(x\right)\nabla q% \right)=\nabla p&\text{in }\Omega\\ q=0&\text{on }\partial\Omega.\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ ( italic_x ) ∇ italic_q ) = ∇ italic_p end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_q = 0 end_CELL start_CELL on ∂ roman_Ω . end_CELL end_ROW end_ARRAY (2.6)

We define w=u+q𝑤𝑢𝑞w=u+qitalic_w = italic_u + italic_q and let  PrV𝑉{}_{V}\ start_FLOATSUBSCRIPT italic_V end_FLOATSUBSCRIPT be the projection operator on V𝑉Vitalic_V. Then, by using (2.6), the system (2.1) can be rewritten under the equivalent form of a diffusion problem with a variable diffusion coefficient

{div(μ(x)w)=f(x)in Ωw=0on Ωcasesdiv𝜇𝑥𝑤𝑓𝑥in Ω𝑤0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu\left(x\right)\nabla w% \right)=f\left(x\right)&\text{in }\Omega\\ w=0&\text{on }\partial\Omega\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ ( italic_x ) ∇ italic_w ) = italic_f ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_w = 0 end_CELL start_CELL on ∂ roman_Ω end_CELL end_ROW end_ARRAY (2.7)

or alternatively

{Lw=f(x)in Ωw=0on Ω.cases𝐿𝑤𝑓𝑥in Ω𝑤0on Ω\left\{\begin{array}[]{ll}Lw=f\left(x\right)&\text{in }\Omega\\ w=0&\text{on }\partial\Omega.\end{array}\right.{ start_ARRAY start_ROW start_CELL italic_L italic_w = italic_f ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_w = 0 end_CELL start_CELL on ∂ roman_Ω . end_CELL end_ROW end_ARRAY (2.8)

Indeed, if wH01(Ω;n)𝑤superscriptsubscript𝐻01Ωsuperscript𝑛w\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right)italic_w ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) solves (2.8), then the pair (u,p)𝑢𝑝\left(u,p\right)( italic_u , italic_p ) where u=𝑢absentu=italic_u =PrwVsubscript𝑤𝑉{}_{V}wstart_FLOATSUBSCRIPT italic_V end_FLOATSUBSCRIPT italic_w and p=L(wu)𝑝𝐿𝑤𝑢\nabla p=L\left(w-u\right)∇ italic_p = italic_L ( italic_w - italic_u ) solves the Stokes system (2.1). Let us notice that the existence and the uniqueness of the solution of problem (2.8) is a direct consequence of the Lax-Milgram theorem.

If in the case of system (2.1), p𝑝\nabla p∇ italic_p can be seen as an intrinsic correction of the external force f𝑓fitalic_f in order to keep the incompressibility of the fluid, in case of system (2.8), we may see the term q𝑞qitalic_q as a correction of the velocity due to the pressure and which is also necessary to express the incompressibility. Thus w=u+q𝑤𝑢𝑞w=u+qitalic_w = italic_u + italic_q can be seen as the recovered velocity under the external force f𝑓fitalic_f and a constant pressure, or equivalently as the density of a diffusion process governed by the source density f.𝑓f.italic_f . As regards the relationship between p𝑝pitalic_p and q,𝑞q,italic_q , namely the equality q=L1p,𝑞superscript𝐿1𝑝q=L^{-1}\nabla p,italic_q = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∇ italic_p , by which the pressure p𝑝pitalic_p is covered into a velocity, we note that q𝑞qitalic_q appears as the solution of the Dirichlet problem (2.6), that is, in terms of diffusion, as a diffusion density due to the source density p.𝑝\nabla p.∇ italic_p .

By associating a diffusion problem to the Stokes model, we are able to compensate the loss in case of the Stokes problem of the maximum principle and of Moser-Harnack type inequalities, some essential tools for the localization and multiplicity of solutions (see [23]), which are the main objectives of our paper. By means of this association, we can however discuss the multiplicity of solutions for the Stokes problem with a reaction force. Let us notice that except for the case of the Navier-Stokes system, where the velocity-dependent convective term (u)u𝑢𝑢(u\cdot\nabla)u( italic_u ⋅ ∇ ) italic_u can be seen as a reaction force, and for the case of the inertial Coriolis force ω×u𝜔𝑢\omega\times uitalic_ω × italic_u (see, e.g., [9], [17]), up to our knowledge, there are few mathematical studies in the literature on Stokes systems with a more general reaction force. In this paper we consider nonlinear reaction forces of the type h(x,u(x)+q(x))𝑥𝑢𝑥𝑞𝑥\ h\left(x,u\left(x\right)+q\left(x\right)\right)italic_h ( italic_x , italic_u ( italic_x ) + italic_q ( italic_x ) ), with q=L1p𝑞superscript𝐿1𝑝q=L^{-1}\nabla pitalic_q = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∇ italic_p. Thus the external force hhitalic_h is sensitive to the recovered velocity w=u+q𝑤𝑢𝑞\ w=u+q\ italic_w = italic_u + italic_q and incorporates both the velocity and the pressure of the Stokes system, by means of the correction function q𝑞qitalic_q (see (2.11)).

Coming back to the association between a diffusion problem and the Stokes system, we remark that if for the reaction-diffusion problem

{Lw=h(x,w(x))in Ωw=0on Ω,cases𝐿𝑤𝑥𝑤𝑥in Ω𝑤0on Ω\left\{\begin{array}[]{ll}Lw=h\left(x,w\left(x\right)\right)&\text{in }\Omega% \\ w=0&\text{on }\partial\Omega,\end{array}\right.{ start_ARRAY start_ROW start_CELL italic_L italic_w = italic_h ( italic_x , italic_w ( italic_x ) ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_w = 0 end_CELL start_CELL on ∂ roman_Ω , end_CELL end_ROW end_ARRAY (2.9)

we are able to localize two solutions w𝑤witalic_w and w¯¯𝑤\overline{w}over¯ start_ARG italic_w end_ARG in the following form

r𝑟\displaystyle ritalic_r \displaystyle\leq w(x)for a.e. xΩ, 0w(x)R for a.e. xΩ;formulae-sequence𝑤𝑥for a.e. 𝑥superscriptΩ 0𝑤𝑥𝑅 for a.e. 𝑥Ω\displaystyle w\left(x\right)\ \ \text{for a.e. }x\in\Omega^{\prime},\ \ 0\leq w% \left(x\right)\leq R\text{ \ for a.e. }x\in\Omega;italic_w ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 0 ≤ italic_w ( italic_x ) ≤ italic_R for a.e. italic_x ∈ roman_Ω ; (2.10)
r¯¯𝑟\displaystyle\overline{r}over¯ start_ARG italic_r end_ARG \displaystyle\leq w¯(x)for a.e. xΩ, 0w¯(x)R¯ for a.e. xΩ,formulae-sequence¯𝑤𝑥for a.e. 𝑥superscriptΩ 0¯𝑤𝑥¯𝑅 for a.e. 𝑥Ω\displaystyle\overline{w}\left(x\right)\ \ \text{for a.e. }x\in\Omega^{\prime}% ,\ \ 0\leq\overline{w}\left(x\right)\leq\overline{R}\text{ \ for a.e. }x\in\Omega,over¯ start_ARG italic_w end_ARG ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 0 ≤ over¯ start_ARG italic_w end_ARG ( italic_x ) ≤ over¯ start_ARG italic_R end_ARG for a.e. italic_x ∈ roman_Ω ,

where ΩΩ\Omega^{\prime}\subset\subset\Omegaroman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ ⊂ roman_Ω is a subdomain of Ω,Ω\Omega,roman_Ω , the vectors r,r¯,R,R¯𝑟¯𝑟𝑅¯𝑅r,\overline{r},R,\overline{R}italic_r , over¯ start_ARG italic_r end_ARG , italic_R , over¯ start_ARG italic_R end_ARG belong to +nsuperscriptsubscript𝑛\mathbb{R}_{+}^{n}blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT with 0<r<Rr¯<R¯,0𝑟𝑅less-than-and-not-equals¯𝑟¯𝑅0<r<R\lvertneqq\overline{r}<\overline{R},0 < italic_r < italic_R ≨ over¯ start_ARG italic_r end_ARG < over¯ start_ARG italic_R end_ARG , and the inequalities are understood component-wise, then the corresponding solutions (u,p),𝑢𝑝\left(u,p\right),( italic_u , italic_p ) , (u¯,p¯)¯𝑢¯𝑝\left(\overline{u},\overline{p}\right)( over¯ start_ARG italic_u end_ARG , over¯ start_ARG italic_p end_ARG ) of the reaction Stokes system

{div(μ(x)u)+p=h(x,u(x)+q(x))in Ωdivu=0in Ωu=0on Ωcasesdiv𝜇𝑥𝑢𝑝𝑥𝑢𝑥𝑞𝑥in Ωdiv𝑢0in Ω𝑢0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu\left(x\right)\nabla u% \right)+\nabla p=h\left(x,u\left(x\right)+q\left(x\right)\right)&\text{in }% \Omega\\ \operatorname{div}\,u=0&\text{in }\Omega\\ u=0&\text{on }\partial\Omega\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ ( italic_x ) ∇ italic_u ) + ∇ italic_p = italic_h ( italic_x , italic_u ( italic_x ) + italic_q ( italic_x ) ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u = 0 end_CELL start_CELL on ∂ roman_Ω end_CELL end_ROW end_ARRAY (2.11)

are distinct. Indeed, if the solutions are not distinct we have w=u+q=u¯+q¯=w¯𝑤𝑢𝑞¯𝑢¯𝑞¯𝑤\ w=u+q=\overline{u}+\overline{q}=\overline{w}italic_w = italic_u + italic_q = over¯ start_ARG italic_u end_ARG + over¯ start_ARG italic_q end_ARG = over¯ start_ARG italic_w end_ARG (u=PrVw,u¯=PrVw¯,q=L1p,q¯=L1p¯),formulae-sequence𝑢subscriptPr𝑉𝑤formulae-sequence¯𝑢subscriptPr𝑉¯𝑤formulae-sequence𝑞superscript𝐿1𝑝¯𝑞superscript𝐿1¯𝑝\left(u=\text{Pr}_{V}w,\,\overline{u}=\text{Pr}_{V}\overline{w},\,q=L^{-1}% \nabla p,\,\overline{q}=L^{-1}\nabla\overline{p}\right),( italic_u = Pr start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT italic_w , over¯ start_ARG italic_u end_ARG = Pr start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT over¯ start_ARG italic_w end_ARG , italic_q = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∇ italic_p , over¯ start_ARG italic_q end_ARG = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∇ over¯ start_ARG italic_p end_ARG ) , which is clearly excluded by the localization relations (2.10), in view of the relation Rr¯.less-than-and-not-equals𝑅¯𝑟\ R\lvertneqq\overline{r}.italic_R ≨ over¯ start_ARG italic_r end_ARG .

In what follows, the localization of solutions will be given in the set DrRsubscript𝐷𝑟𝑅D_{rR}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT defined by

DrR={wH01(Ω;n):rw(x)for a.e. xΩ, 0w(x)R for a.e. xΩ}.subscript𝐷𝑟𝑅conditional-set𝑤superscriptsubscript𝐻01Ωsuperscript𝑛formulae-sequence𝑟𝑤𝑥formulae-sequencefor a.e. 𝑥superscriptΩ 0𝑤𝑥𝑅 for a.e. 𝑥ΩD_{rR}=\left\{w\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right):\ r\leq w\left(% x\right)\ \ \text{for a.e. }x\in\Omega^{\prime},\ \ 0\leq w\left(x\right)\leq R% \text{ \ for a.e. }x\in\Omega\right\}.italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT = { italic_w ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) : italic_r ≤ italic_w ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 0 ≤ italic_w ( italic_x ) ≤ italic_R for a.e. italic_x ∈ roman_Ω } . (2.12)

For i{1,..,n},i\in\left\{1,..,n\right\},italic_i ∈ { 1 , . . , italic_n } , we shall also use the notation

DriRi={wiH01(Ω):riwi(x)for a.e. xΩ, 0wi(x)Ri for a.e. xΩ}.subscript𝐷subscript𝑟𝑖subscript𝑅𝑖conditional-setsubscript𝑤𝑖superscriptsubscript𝐻01Ωformulae-sequencesubscript𝑟𝑖subscript𝑤𝑖𝑥formulae-sequencefor a.e. 𝑥superscriptΩ 0subscript𝑤𝑖𝑥subscript𝑅𝑖 for a.e. 𝑥ΩD_{r_{i}R_{i}}=\left\{w_{i}\in H_{0}^{1}\left(\Omega\right):\ r_{i}\leq w_{i}% \left(x\right)\ \ \text{for a.e. }x\in\Omega^{\prime},\ \ 0\leq w_{i}\left(x% \right)\leq R_{i}\text{ \ for a.e. }x\in\Omega\right\}.italic_D start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = { italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) : italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 0 ≤ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ≤ italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for a.e. italic_x ∈ roman_Ω } .

Notice that in general system (2.9) does not have a variational form. However, each of its equations has one. Thus we may consider the partial energy functionals of (2.9), namely the functionals Ji:H01(Ω;n)(i=1,..,n),\ J_{i}:H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right)\rightarrow\mathbb{R}\ \ % \left(i=1,..,n\right),italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) → blackboard_R ( italic_i = 1 , . . , italic_n ) , defined by

Ji(w)=Ω(12μ(x)|wi|20wi(x)hi(x,w1(x),..,wi1(x),τi,wi+1(x),..,wn(x))dτi)dx,J_{i}\left(w\right)=\int_{\Omega}\left(\frac{1}{2}\mu\left(x\right)\left|% \nabla w_{i}\right|^{2}-\int_{0}^{w_{i}\left(x\right)}h_{i}\left(x,w_{1}\left(% x\right),..,w_{i-1}\left(x\right),\tau_{i},w_{i+1}\left(x\right),..,w_{n}\left% (x\right)\right)d\tau_{i}\right)dx,italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_w ) = ∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_μ ( italic_x ) | ∇ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) end_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ) , . . , italic_w start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT ( italic_x ) , italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT ( italic_x ) , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) ) italic_d italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_d italic_x ,

which will allow to characterize the solution w𝑤witalic_w of the stationary model (2.9) in terms of Nash equilibrium.

To end this introductory section we recall some basic results from the theory of linear elliptic equations and the theory of matrices.

(R1) If f(x)0𝑓𝑥0f(x)\geq 0italic_f ( italic_x ) ≥ 0 for a.e. xΩ𝑥Ωx\in\Omegaitalic_x ∈ roman_Ω, then the weak maximum principle (see for instance [13, Theorem 8.1]) guarantees that the solution w=L1f𝑤superscript𝐿1𝑓w=L^{-1}fitalic_w = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f of problem (2.8) satisfies w(x)0𝑤𝑥0w\left(x\right)\geq 0italic_w ( italic_x ) ≥ 0 a.e. in Ω.Ω\Omega.roman_Ω .

(R2) If fL(Ω)𝑓superscript𝐿Ωf\in L^{\infty}(\Omega)italic_f ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω ) then, according to [13, Theorem 8.15], the solution w𝑤witalic_w of problem (2.8) belongs to L(Ω)superscript𝐿ΩL^{\infty}(\Omega)italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω ) and there is a constant ΓfsubscriptΓ𝑓\Gamma_{f}roman_Γ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT independent of μ𝜇\muitalic_μ which satisfies 0<μ¯μ(x)μ¯,0¯𝜇𝜇𝑥¯𝜇0<\underline{\mu}\leq\mu\left(x\right)\leq\overline{\mu},0 < under¯ start_ARG italic_μ end_ARG ≤ italic_μ ( italic_x ) ≤ over¯ start_ARG italic_μ end_ARG , such that

wL(Ω)Γf.subscriptnorm𝑤superscript𝐿ΩsubscriptΓ𝑓\left\|w\right\|_{L^{\infty}(\Omega)}\,\leq\Gamma_{f}.∥ italic_w ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ≤ roman_Γ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT .

(R3) For each fL(Ω)𝑓superscript𝐿Ωf\in L^{\infty}\left(\Omega\right)italic_f ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω ) with f0𝑓0f\geq 0italic_f ≥ 0 in ΩΩ\Omegaroman_Ω and f>0𝑓0f>0italic_f > 0 on a subset of nonzero measure of Ω,Ω\Omega,roman_Ω , there exists γf>0subscript𝛾𝑓0\gamma_{f}>0italic_γ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT > 0 such that the solution w𝑤witalic_w of problem (2.8) verifies w(x)γf𝑤𝑥subscript𝛾𝑓\ w\left(x\right)\geq\gamma_{f}\ italic_w ( italic_x ) ≥ italic_γ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT for a.e. x𝑥xitalic_x in the fixed subdomain ΩΩ.\Omega^{\prime}\subset\subset\Omega.roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ ⊂ roman_Ω . For μ𝜇\muitalic_μ which satisfies 0<μ¯μ(x)μ¯,0¯𝜇𝜇𝑥¯𝜇0<\underline{\mu}\leq\mu\left(x\right)\leq\overline{\mu},0 < under¯ start_ARG italic_μ end_ARG ≤ italic_μ ( italic_x ) ≤ over¯ start_ARG italic_μ end_ARG , the constant γfsubscript𝛾𝑓\ \gamma_{f}italic_γ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT   can be chosen to be independent of μ𝜇\muitalic_μ (see [15, Theorem 12.1.2]).

For a square matrix \mathcal{M}caligraphic_M with nonnegative entries, the spectral radius ρ()𝜌\rho\left(\mathcal{M}\right)italic_ρ ( caligraphic_M ) is the maximum of the modulus of its eigenvalues λ,𝜆\lambda,italic_λ , i.e., of all λ𝜆\lambda\in\mathbb{C}italic_λ ∈ blackboard_C with det(λI)=0.𝜆𝐼0\ \det\left(\mathcal{M}-\lambda I\right)=0.roman_det ( caligraphic_M - italic_λ italic_I ) = 0 . There are known the following characterizations of such kind of matrices:

(a) ksuperscript𝑘\mathcal{M}^{k}caligraphic_M start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT tends to the zero matrix as k;𝑘k\rightarrow\infty;italic_k → ∞ ;

(b) The matrix I𝐼I-\mathcal{M}italic_I - caligraphic_M is invertible and the entries of (I)1superscript𝐼1\left(I-\mathcal{M}\right)^{-1}( italic_I - caligraphic_M ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT are also nonnegative.

3 Main results

We give now the main results of the paper. In Section 3.1, the localization and multiplicity results are proven for the recovered velocity of system (2.11), under the hypotheses (𝐡𝟏)𝐡𝟏\mathbf{(h1)}( bold_h1 ) and (𝐡𝟐)𝐡𝟐\mathbf{(h2)}( bold_h2 ) below on the nonlinear right-hand side data function. In Section 3.2 we deal with the case of a uniform pressure. In this case, the recovered velocity and the velocity of the Stokes system coincide and thus the localization and the multiplicity results are obtained for the velocity itself. We first give the sufficient condition (3.10) in order the pressure of a Stokes system to be uniform and then we prove localization results for the system (3.11), under the hypotheses (3.16) on the right-hand side data function. These latest results are proven in the planar case.

3.1 Localization of the recovered velocity

We start by giving a localization of the solution for the linear diffusion problem (2.8), which for the classical Stokes system (2.1) represents a localization of the recovered velocity u+q𝑢𝑞u+qitalic_u + italic_q in terms of some bounds of the external force f(x).𝑓𝑥f\left(x\right).italic_f ( italic_x ) .

Proposition 1

If

rγ1Ωf(x)for a.e. xΩand 0f(x)RΓ1for a.e. xΩ,formulae-sequenceformulae-sequence𝑟subscript𝛾subscript1superscriptΩ𝑓𝑥formulae-sequencefor a.e. 𝑥superscriptΩand 0𝑓𝑥𝑅subscriptΓ1for a.e. 𝑥Ω\frac{r}{\gamma_{1_{\Omega^{\prime}}}}\leq f\left(x\right)\ \ \ \text{for a.e.% }x\in\Omega^{\prime}\ \ \ \text{and\ \ \ }0\leq f\left(x\right)\leq\frac{R}{% \Gamma_{1}}\ \ \text{for a.e. }x\in\Omega,divide start_ARG italic_r end_ARG start_ARG italic_γ start_POSTSUBSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG ≤ italic_f ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and 0 ≤ italic_f ( italic_x ) ≤ divide start_ARG italic_R end_ARG start_ARG roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG for a.e. italic_x ∈ roman_Ω ,

then the solution w=L1f𝑤superscript𝐿1𝑓\ w=L^{-1}fitalic_w = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f   of (2.8) satisfies

rw(x)for a.e. xΩand 0w(x)Rfor a.e. xΩ.formulae-sequenceformulae-sequence𝑟𝑤𝑥formulae-sequencefor a.e. 𝑥superscriptΩand 0𝑤𝑥𝑅for a.e. 𝑥Ωr\leq w\left(x\right)\ \ \ \text{for a.e. }x\in\Omega^{\prime}\ \ \ \text{and% \ \ \ }0\leq w\left(x\right)\leq R\ \ \text{for a.e. }x\in\Omega.italic_r ≤ italic_w ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and 0 ≤ italic_w ( italic_x ) ≤ italic_R for a.e. italic_x ∈ roman_Ω .

Here 1Ωsubscript1superscriptΩ1_{\Omega^{\prime}}1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is the characteristic function of the domain ΩsuperscriptΩ\ \Omega^{\prime}roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and γ1Ω,Γ1subscript𝛾subscript1superscriptΩsubscriptΓ1\gamma_{1_{\Omega^{\prime}}},\ \Gamma_{1}italic_γ start_POSTSUBSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT , roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT are the constants given by the remarks (R3), (R2), associated to the functions 1Ωsubscript1superscriptΩ1_{\Omega^{\prime}}1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT and  constant 1,11,1 , respectively.

Proof. From the relation f(x)0𝑓𝑥0f\left(x\right)\geq 0italic_f ( italic_x ) ≥ 0 on Ω,Ω\Omega,roman_Ω , according to (R1), one has w0𝑤0w\geq 0italic_w ≥ 0 on Ω.Ω\Omega.roman_Ω . Next, using (R2), from the relation f(x)RΓ1𝑓𝑥𝑅subscriptΓ1f\left(x\right)\leq\frac{R}{\Gamma_{1}}italic_f ( italic_x ) ≤ divide start_ARG italic_R end_ARG start_ARG roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG on Ω,Ω\Omega,roman_Ω , we have

w(x)RΓ1(L11)(x)RΓ1Γ1=Ron Ω.formulae-sequence𝑤𝑥𝑅subscriptΓ1superscript𝐿11𝑥𝑅subscriptΓ1subscriptΓ1𝑅on Ωw\left(x\right)\leq\frac{R}{\Gamma_{1}}\left(L^{-1}1\right)\left(x\right)\leq% \frac{R}{\Gamma_{1}}\Gamma_{1}=R\ \ \text{on }\Omega.italic_w ( italic_x ) ≤ divide start_ARG italic_R end_ARG start_ARG roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ( italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT 1 ) ( italic_x ) ≤ divide start_ARG italic_R end_ARG start_ARG roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_R on roman_Ω .

Similarly, using (R3), since f(x)rγ1Ω1Ω(x)𝑓𝑥𝑟subscript𝛾subscript1superscriptΩsubscript1superscriptΩ𝑥\displaystyle f\left(x\right)\geq\frac{r}{\gamma_{1_{\Omega^{\prime}}}}1_{% \Omega^{\prime}}\left(x\right)italic_f ( italic_x ) ≥ divide start_ARG italic_r end_ARG start_ARG italic_γ start_POSTSUBSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) on Ω,Ω\Omega,roman_Ω , we obtain

w(x)rγ1Ω(L11Ω)(x)rγ1Ωγ1Ω=ron Ω.formulae-sequence𝑤𝑥𝑟subscript𝛾subscript1superscriptΩsuperscript𝐿1subscript1superscriptΩ𝑥𝑟subscript𝛾subscript1superscriptΩsubscript𝛾subscript1superscriptΩ𝑟on superscriptΩw\left(x\right)\geq\frac{r}{\gamma_{1_{\Omega^{\prime}}}}\left(L^{-1}1_{\Omega% ^{\prime}}\right)\left(x\right)\geq\frac{r}{\gamma_{1_{\Omega^{\prime}}}}% \gamma_{1_{\Omega^{\prime}}}=r\ \ \text{on }\Omega^{\prime}.italic_w ( italic_x ) ≥ divide start_ARG italic_r end_ARG start_ARG italic_γ start_POSTSUBSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG ( italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ( italic_x ) ≥ divide start_ARG italic_r end_ARG start_ARG italic_γ start_POSTSUBSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG italic_γ start_POSTSUBSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_r on roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

 

We now state and prove an existence and localization theorem for the semi-linear reaction-diffusion problem (2.9) which consequently leads to an existence and localization result for the reaction Stokes system (2.11). The assumptions on the data are as follows:

(h1)

There are two functions h¯,h¯L(Ω;+n)¯¯superscript𝐿Ωsuperscriptsubscript𝑛\ \underline{h},\ \overline{h}\in L^{\infty}\left(\Omega;\mathbb{R}_{+}^{n}\right)under¯ start_ARG italic_h end_ARG , over¯ start_ARG italic_h end_ARG ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) such that

h¯(x)¯𝑥\displaystyle\underline{h}\left(x\right)under¯ start_ARG italic_h end_ARG ( italic_x ) \displaystyle\leq h(x,τ)h¯(x)on Ω×[r,R],𝑥𝜏¯𝑥on superscriptΩ𝑟𝑅\displaystyle h\left(x,\tau\right)\leq\overline{h}\left(x\right)\ \ \text{on % \ }\Omega^{\prime}\times\left[r,R\right],italic_h ( italic_x , italic_τ ) ≤ over¯ start_ARG italic_h end_ARG ( italic_x ) on roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT × [ italic_r , italic_R ] , (3.1)
h¯(x)¯𝑥\displaystyle\underline{h}\left(x\right)under¯ start_ARG italic_h end_ARG ( italic_x ) \displaystyle\leq h(x,τ)=h(x)h¯(x)on (ΩΩ)×+n,formulae-sequence𝑥𝜏𝑥¯𝑥on ΩsuperscriptΩsuperscriptsubscript𝑛\displaystyle h\left(x,\tau\right)=h\left(x\right)\leq\overline{h}\left(x% \right)\ \ \text{on \ }\left(\Omega\setminus\Omega^{\prime}\right)\times% \mathbb{R}_{+}^{n},italic_h ( italic_x , italic_τ ) = italic_h ( italic_x ) ≤ over¯ start_ARG italic_h end_ARG ( italic_x ) on ( roman_Ω ∖ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ,
rL1h¯ on Ωand L1h¯Ron Ω.formulae-sequence𝑟superscript𝐿1¯ on superscriptΩand superscript𝐿1¯𝑅on Ωr\leq L^{-1}\underline{h}\text{ \ \ on\ }\Omega^{\prime}\ \ \ \text{and\ \ \ }% L^{-1}\overline{h}\leq R\ \ \ \text{on\ }\Omega.italic_r ≤ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT under¯ start_ARG italic_h end_ARG on roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over¯ start_ARG italic_h end_ARG ≤ italic_R on roman_Ω .
(h2)

There exist constants lij0subscript𝑙𝑖𝑗0\ l_{ij}\geq 0italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ≥ 0 (i,j=1,..,n)\left(i,j=1,..,n\right)( italic_i , italic_j = 1 , . . , italic_n ) such that for each i{1,..,n},i\in\left\{1,..,n\right\},italic_i ∈ { 1 , . . , italic_n } ,

|hi(x,τ)hi(x,τ¯)|j=1nlij|τjτ¯j|for a.e. xΩ and all τ,τ¯[r,R]formulae-sequencesubscript𝑖𝑥𝜏subscript𝑖𝑥¯𝜏superscriptsubscript𝑗1𝑛subscript𝑙𝑖𝑗subscript𝜏𝑗subscript¯𝜏𝑗formulae-sequencefor a.e. 𝑥superscriptΩ and all 𝜏¯𝜏𝑟𝑅\left|h_{i}\left(x,\tau\right)-h_{i}\left(x,\overline{\tau}\right)\right|\leq% \sum\limits_{j=1}^{n}l_{ij}\left|\tau_{j}-\overline{\tau}_{j}\right|\ \ \ % \text{for a.e. }x\in\Omega^{\prime}\text{ and all }\tau,\overline{\tau}\in\left[r,R\right]| italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_τ ) - italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , over¯ start_ARG italic_τ end_ARG ) | ≤ ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT | italic_τ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - over¯ start_ARG italic_τ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT | for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and all italic_τ , over¯ start_ARG italic_τ end_ARG ∈ [ italic_r , italic_R ] (3.2)

and the spectral radius of the matrix

=(Lij)1i,jn=(lijμ¯λ1)1i,jnsubscriptsubscript𝐿𝑖𝑗formulae-sequence1𝑖𝑗𝑛subscriptsubscript𝑙𝑖𝑗¯𝜇subscript𝜆1formulae-sequence1𝑖𝑗𝑛\mathcal{M}=\left(L_{ij}\right)_{1\leq i,j\leq n}=\left(\frac{l_{ij}}{% \underline{\mu}\lambda_{1}}\right)_{1\leq i,j\leq n}caligraphic_M = ( italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 1 ≤ italic_i , italic_j ≤ italic_n end_POSTSUBSCRIPT = ( divide start_ARG italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG start_ARG under¯ start_ARG italic_μ end_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUBSCRIPT 1 ≤ italic_i , italic_j ≤ italic_n end_POSTSUBSCRIPT

is strictly less than one. Here λ1subscript𝜆1\lambda_{1}italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the first eigenvalue of the Dirichlet problem for the operator Δ.Δ-\Delta.- roman_Δ .

Remark 2

(a) Condition (3.1), more exactly the requirement h(x,τ)=h(x)𝑥𝜏𝑥\ h\left(x,\tau\right)=h\left(x\right)italic_h ( italic_x , italic_τ ) = italic_h ( italic_x ) on (ΩΩ)×+nΩsuperscriptΩsuperscriptsubscript𝑛\left(\Omega\setminus\Omega^{\prime}\right)\times\mathbb{R}_{+}^{n}( roman_Ω ∖ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) × blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT shows that the external force does not react in the exterior of Ω,superscriptΩ\Omega^{\prime},roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , namely in the vicinity of the boundary Ω.Ω\partial\Omega.∂ roman_Ω . For example, h(x,τ)=f0(x)+g(x)f(τ)𝑥𝜏subscript𝑓0𝑥𝑔𝑥𝑓𝜏\ h\left(x,\tau\right)=f_{0}\left(x\right)+g\left(x\right)f\left(\tau\right)italic_h ( italic_x , italic_τ ) = italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_x ) + italic_g ( italic_x ) italic_f ( italic_τ ) with g(x)=0𝑔𝑥0\ g\left(x\right)=0italic_g ( italic_x ) = 0 on ΩΩΩsuperscriptΩ\Omega\setminus\Omega^{\prime}roman_Ω ∖ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is such a function.

(b) We underline the local form of the conditions imposed on h,h,italic_h , which are given only on a vector interval [r,R].𝑟𝑅\left[r,R\right].[ italic_r , italic_R ] . This makes possible for a given function hhitalic_h these conditions to be satisfied for several intervals [r,R]𝑟𝑅\left[r,R\right][ italic_r , italic_R ] leading then to multiple solutions.

Theorem 3

Assume that conditions (h1) and (h2) hold. Then problem (2.9) has a unique solution w𝑤witalic_w in DrR,subscript𝐷𝑟𝑅D_{rR},italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT , with DrRsubscript𝐷𝑟𝑅D_{rR}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT defined in (2.12). Moreover, the solution w𝑤witalic_w is in DrRsubscript𝐷𝑟𝑅D_{rR}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT a Nash equilibrium with respect to the partial energy functionals of the system.

Proof. (a) Existence, uniqueness and localization. Let us notice that system (2.9) is equivalent to the fixed point equation w=N(w),wDrR,formulae-sequence𝑤𝑁𝑤𝑤subscript𝐷𝑟𝑅\ w=N\left(w\right),\ w\in D_{rR},italic_w = italic_N ( italic_w ) , italic_w ∈ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT , where N(w)=L1h(,w()).𝑁𝑤superscript𝐿1𝑤\ N\left(w\right)=L^{-1}h\left(\cdot,w\left(\cdot\right)\right).italic_N ( italic_w ) = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h ( ⋅ , italic_w ( ⋅ ) ) . We apply Perov’s vector version of Banach’s fixed point theorem. Perov’s vectorial approach, initiated by Perov ([21]) in connection with the contraction principle, was extended for instance in [22] and [3] for other results from nonlinear functional analysis. We first prove the invariance condition N(DrR)DrR.𝑁subscript𝐷𝑟𝑅subscript𝐷𝑟𝑅\ N\left(D_{rR}\right)\subset D_{rR}.italic_N ( italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT ) ⊂ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT . To this aim, let w𝑤witalic_w be any element of DrR.subscript𝐷𝑟𝑅D_{rR}.italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT . Using (h1) we have

h¯(x)h(x,w(x))h¯(x)for a.e. xΩ,formulae-sequence¯𝑥𝑥𝑤𝑥¯𝑥for a.e. 𝑥Ω\underline{h}\left(x\right)\leq h\left(x,w\left(x\right)\right)\leq\overline{h% }\left(x\right)\ \ \ \text{for a.e. }x\in\Omega,under¯ start_ARG italic_h end_ARG ( italic_x ) ≤ italic_h ( italic_x , italic_w ( italic_x ) ) ≤ over¯ start_ARG italic_h end_ARG ( italic_x ) for a.e. italic_x ∈ roman_Ω ,

whence

00\displaystyle 0 \displaystyle\leq (L1h¯)(x)N(w)(x)(L1h¯)(x)Ron Ω,formulae-sequencesuperscript𝐿1¯𝑥𝑁𝑤𝑥superscript𝐿1¯𝑥𝑅on Ω\displaystyle\left(L^{-1}\underline{h}\right)\left(x\right)\leq N\left(w\right% )\left(x\right)\leq\left(L^{-1}\overline{h}\right)\left(x\right)\leq R\ \ \ % \text{on\ }\Omega,( italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT under¯ start_ARG italic_h end_ARG ) ( italic_x ) ≤ italic_N ( italic_w ) ( italic_x ) ≤ ( italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over¯ start_ARG italic_h end_ARG ) ( italic_x ) ≤ italic_R on roman_Ω ,
N(w)(x)𝑁𝑤𝑥\displaystyle N\left(w\right)\left(x\right)italic_N ( italic_w ) ( italic_x ) \displaystyle\geq (L1h¯)(x)ron Ω.superscript𝐿1¯𝑥𝑟on superscriptΩ\displaystyle\left(L^{-1}\underline{h}\right)\left(x\right)\geq r\ \ \ \text{% on }\Omega^{\prime}.( italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT under¯ start_ARG italic_h end_ARG ) ( italic_x ) ≥ italic_r on roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

Hence N(w)DrR.𝑁𝑤subscript𝐷𝑟𝑅\ N\left(w\right)\in D_{rR}.italic_N ( italic_w ) ∈ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT .

Next we prove that N𝑁Nitalic_N is a Perov contraction on DrR.subscript𝐷𝑟𝑅D_{rR}.italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT . To show this, let w1,w2DrRsuperscript𝑤1superscript𝑤2subscript𝐷𝑟𝑅w^{1},w^{2}\in D_{rR}italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∈ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT and denote vi=N(wi),superscript𝑣𝑖𝑁superscript𝑤𝑖v^{i}=N\left(w^{i}\right),italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_N ( italic_w start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) , i=1,2.𝑖12i=1,2.italic_i = 1 , 2 . Then one has

{L(v1v2)=h(x,w1)h(x,w2)in Ωv1v2=0on Ωcases𝐿superscript𝑣1superscript𝑣2𝑥superscript𝑤1𝑥superscript𝑤2in Ωsuperscript𝑣1superscript𝑣20on Ω\left\{\begin{array}[]{ll}L\left(v^{1}-v^{2}\right)=h\left(x,w^{1}\right)-h% \left(x,w^{2}\right)&\text{in }\Omega\\ v^{1}-v^{2}=0&\text{on }\partial\Omega\end{array}\right.{ start_ARRAY start_ROW start_CELL italic_L ( italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_h ( italic_x , italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_h ( italic_x , italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω end_CELL end_ROW end_ARRAY

in the weak sense. Thus, multiplying the i𝑖iitalic_i-th equation by vi1vi2superscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2v_{i}^{1}-v_{i}^{2}italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and integrating over ΩΩ\Omegaroman_Ω yield

Ωμ(x)(vi1vi2)(vi1vi2)dx=Ω(hi(x,w1)hi(x,w2))(vi1vi2)𝑑x.subscriptΩ𝜇𝑥superscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2superscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2𝑑𝑥subscriptΩsubscript𝑖𝑥superscript𝑤1subscript𝑖𝑥superscript𝑤2superscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2differential-d𝑥\int_{\Omega}\mu\left(x\right)\nabla\left(v_{i}^{1}-v_{i}^{2}\right)\cdot% \nabla\left(v_{i}^{1}-v_{i}^{2}\right)\,dx=\int_{\Omega}\left(h_{i}\left(x,w^{% 1}\right)-h_{i}\left(x,w^{2}\right)\right)\left(v_{i}^{1}-v_{i}^{2}\right)\,dx.∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT italic_μ ( italic_x ) ∇ ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ⋅ ∇ ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_d italic_x = ∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_d italic_x . (3.3)

The left-hand side is greater than μ¯vi1vi2H01(Ω)2,¯𝜇superscriptsubscriptnormsuperscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2superscriptsubscript𝐻01Ω2\underline{\mu}\left\|v_{i}^{1}-v_{i}^{2}\right\|_{H_{0}^{1}\left(\Omega\right% )}^{2},under¯ start_ARG italic_μ end_ARG ∥ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , while for the right-hand side, in view of the fact that h(x,τ)𝑥𝜏h\left(x,\tau\right)italic_h ( italic_x , italic_τ ) does not depend on τ𝜏\tauitalic_τ for xΩΩ𝑥ΩsuperscriptΩx\in\Omega\setminus\Omega^{\prime}italic_x ∈ roman_Ω ∖ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and using (h2), we have

Ω(hi(x,w1)hi(x,w2))(vi1vi2)𝑑xsubscriptΩsubscript𝑖𝑥superscript𝑤1subscript𝑖𝑥superscript𝑤2superscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2differential-d𝑥\displaystyle\int_{\Omega}\left(h_{i}\left(x,w^{1}\right)-h_{i}\left(x,w^{2}% \right)\right)\left(v_{i}^{1}-v_{i}^{2}\right)\,dx∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_d italic_x =\displaystyle== Ω(hi(x,w1)hi(x,w2))(vi1vi2)𝑑xsubscriptsuperscriptΩsubscript𝑖𝑥superscript𝑤1subscript𝑖𝑥superscript𝑤2superscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2differential-d𝑥\displaystyle\int_{\Omega^{\prime}}\left(h_{i}\left(x,w^{1}\right)-h_{i}\left(% x,w^{2}\right)\right)\left(v_{i}^{1}-v_{i}^{2}\right)\,dx∫ start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x , italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_d italic_x
\displaystyle\leq j=1nlijΩ|wj1wj2||vi1vi2|superscriptsubscript𝑗1𝑛subscript𝑙𝑖𝑗subscriptsuperscriptΩsuperscriptsubscript𝑤𝑗1superscriptsubscript𝑤𝑗2superscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2\displaystyle\sum\limits_{j=1}^{n}l_{ij}\int_{\Omega^{\prime}}\left|w_{j}^{1}-% w_{j}^{2}\right|\left|v_{i}^{1}-v_{i}^{2}\right|∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | | italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT |
\displaystyle\leq j=1nlijwj1wj2L2(Ω)vi1vi2L2(Ω)superscriptsubscript𝑗1𝑛subscript𝑙𝑖𝑗subscriptnormsuperscriptsubscript𝑤𝑗1superscriptsubscript𝑤𝑗2superscript𝐿2Ωsubscriptnormsuperscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2superscript𝐿2Ω\displaystyle\sum\limits_{j=1}^{n}l_{ij}\left\|w_{j}^{1}-w_{j}^{2}\right\|_{L^% {2}\left(\Omega\right)}\left\|v_{i}^{1}-v_{i}^{2}\right\|_{L^{2}\left(\Omega% \right)}∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ∥ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT
\displaystyle\leq 1λ1(j=1nlijwj1wj2H01(Ω))vi1vi2H01(Ω),1subscript𝜆1superscriptsubscript𝑗1𝑛subscript𝑙𝑖𝑗subscriptnormsuperscriptsubscript𝑤𝑗1superscriptsubscript𝑤𝑗2superscriptsubscript𝐻01Ωsubscriptnormsuperscriptsubscript𝑣𝑖1superscriptsubscript𝑣𝑖2superscriptsubscript𝐻01Ω\displaystyle\frac{1}{\lambda_{1}}\left(\sum\limits_{j=1}^{n}l_{ij}\left\|w_{j% }^{1}-w_{j}^{2}\right\|_{H_{0}^{1}\left(\Omega\right)}\right)\left\|v_{i}^{1}-% v_{i}^{2}\right\|_{H_{0}^{1}\left(\Omega\right)},divide start_ARG 1 end_ARG start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ( ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ) ∥ italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ,

where for the last inequality Poincaré’s inequality with the best constant 1/λ11subscript𝜆11/\sqrt{\lambda_{1}}1 / square-root start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG has been applied twice. Hence

Ni(w1)Ni(w2)H01(Ω)1μ¯λ1j=1nlijwj1wj2H01(Ω)(i=1,..,n).\left\|N_{i}\left(w^{1}\right)-N_{i}\left(w^{2}\right)\right\|_{H_{0}^{1}\left% (\Omega\right)}\leq\frac{1}{\underline{\mu}\lambda_{1}}\sum\limits_{j=1}^{n}l_% {ij}\left\|w_{j}^{1}-w_{j}^{2}\right\|_{H_{0}^{1}\left(\Omega\right)}\ \ \ % \left(i=1,..,n\right).∥ italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG under¯ start_ARG italic_μ end_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ( italic_i = 1 , . . , italic_n ) .

These n𝑛nitalic_n inequalities can be put together under the form of the matrix inequality

(N1(w1)N1(w2)H01(Ω)..Nn(w1)Nn(w2)H01(Ω))(l11μ¯λ1..l1nμ¯λ1......ln1μ¯λ1..lnnμ¯λ1)(w11w12H01(Ω)..wn1wn2H01(Ω))subscriptnormsubscript𝑁1superscript𝑤1subscript𝑁1superscript𝑤2superscriptsubscript𝐻01Ωabsentsubscriptnormsubscript𝑁𝑛superscript𝑤1subscript𝑁𝑛superscript𝑤2superscriptsubscript𝐻01Ωsubscript𝑙11¯𝜇subscript𝜆1absentsubscript𝑙1𝑛¯𝜇subscript𝜆1absentabsentabsentsubscript𝑙𝑛1¯𝜇subscript𝜆1absentsubscript𝑙𝑛𝑛¯𝜇subscript𝜆1subscriptnormsuperscriptsubscript𝑤11superscriptsubscript𝑤12superscriptsubscript𝐻01Ωabsentsubscriptnormsuperscriptsubscript𝑤𝑛1superscriptsubscript𝑤𝑛2superscriptsubscript𝐻01Ω\left(\begin{array}[]{c}\left\|N_{1}\left(w^{1}\right)-N_{1}\left(w^{2}\right)% \right\|_{H_{0}^{1}\left(\Omega\right)}\\ ..\\ \left\|N_{n}\left(w^{1}\right)-N_{n}\left(w^{2}\right)\right\|_{H_{0}^{1}\left% (\Omega\right)}\end{array}\right)\leq\left(\begin{array}[]{ccc}\frac{l_{11}}{% \underline{\mu}\lambda_{1}}&..&\frac{l_{1n}}{\underline{\mu}\lambda_{1}}\\ ..&..&..\\ \frac{l_{n1}}{\underline{\mu}\lambda_{1}}&..&\frac{l_{nn}}{\underline{\mu}% \lambda_{1}}\end{array}\right)\left(\begin{array}[]{c}\left\|w_{1}^{1}-w_{1}^{% 2}\right\|_{H_{0}^{1}\left(\Omega\right)}\\ ..\\ \left\|w_{n}^{1}-w_{n}^{2}\right\|_{H_{0}^{1}\left(\Omega\right)}\end{array}\right)( start_ARRAY start_ROW start_CELL ∥ italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL ∥ italic_N start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_N start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) ≤ ( start_ARRAY start_ROW start_CELL divide start_ARG italic_l start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_ARG start_ARG under¯ start_ARG italic_μ end_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_CELL start_CELL . . end_CELL start_CELL divide start_ARG italic_l start_POSTSUBSCRIPT 1 italic_n end_POSTSUBSCRIPT end_ARG start_ARG under¯ start_ARG italic_μ end_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL . . end_CELL start_CELL . . end_CELL start_CELL . . end_CELL end_ROW start_ROW start_CELL divide start_ARG italic_l start_POSTSUBSCRIPT italic_n 1 end_POSTSUBSCRIPT end_ARG start_ARG under¯ start_ARG italic_μ end_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_CELL start_CELL . . end_CELL start_CELL divide start_ARG italic_l start_POSTSUBSCRIPT italic_n italic_n end_POSTSUBSCRIPT end_ARG start_ARG under¯ start_ARG italic_μ end_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW end_ARRAY ) ( start_ARRAY start_ROW start_CELL ∥ italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL ∥ italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY )

where the spectral radius of the matrix (lij/(μ¯λ1))1i,jnsubscriptsubscript𝑙𝑖𝑗¯𝜇subscript𝜆1formulae-sequence1𝑖𝑗𝑛\left(l_{ij}/\left(\underline{\mu}\lambda_{1}\right)\right)_{1\leq i,j\leq n}( italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT / ( under¯ start_ARG italic_μ end_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT 1 ≤ italic_i , italic_j ≤ italic_n end_POSTSUBSCRIPT is less than one. Thus N𝑁Nitalic_N is a Perov contraction on DrR.subscript𝐷𝑟𝑅D_{rR}.italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT . This implies that N𝑁Nitalic_N has a unique fixed point w𝑤witalic_w in DrR.subscript𝐷𝑟𝑅D_{rR}.italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT .

(b) Nash equilibrium. In order to prove that the solution w=(w1,..,wn)\ w=\left(w_{1},..,w_{n}\right)\ italic_w = ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is a Nash equilibrium, we use an iterative approximation scheme. We start with some fixed elements w20,..,wn0w_{2}^{0},..,w_{n}^{0}italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT in Dr2R2×..×DrnRn.D_{r_{2}R_{2}}\times..\times D_{r_{n}R_{n}}.italic_D start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT × . . × italic_D start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT . At each step k𝑘kitalic_k (k1𝑘1k\geq 1italic_k ≥ 1), the elements w2k1,..,wnk1w_{2}^{k-1},..,w_{n}^{k-1}italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT having been determined at step k1𝑘1k-1italic_k - 1, first we apply Ekeland’s principle to J1(,w2k1,..,wnk1)\ J_{1}\left(\cdot\ ,w_{2}^{k-1},..,w_{n}^{k-1}\right)italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( ⋅ , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) and find a w1kDr1R1superscriptsubscript𝑤1𝑘subscript𝐷subscript𝑟1subscript𝑅1\ w_{1}^{k}\in D_{r_{1}R_{1}}italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ italic_D start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT such that

J1(w1k,w2k1,..,wnk1)infDr1R1J1(,w2k1,..,wnk1)+1k,J11(w1k,w2k1,..,wnk1)H01(Ω)1k.J_{1}\left(w_{1}^{k},w_{2}^{k-1},..,w_{n}^{k-1}\right)\leq\inf_{D_{r_{1}R_{1}}% }J_{1}\left(\cdot,w_{2}^{k-1},..,w_{n}^{k-1}\right)+\frac{1}{k},\ \ \left\|J_{% 11}\left(w_{1}^{k},w_{2}^{k-1},..,w_{n}^{k-1}\right)\right\|_{H_{0}^{1}\left(% \Omega\right)}\leq\frac{1}{k}.italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) ≤ roman_inf start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( ⋅ , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) + divide start_ARG 1 end_ARG start_ARG italic_k end_ARG , ∥ italic_J start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG italic_k end_ARG .

Next we apply Ekeland’s principle to J2(w1k,,w3k1,..,wnk1)J_{2}\left(w_{1}^{k},\cdot\ ,w_{3}^{k-1},..,w_{n}^{k-1}\right)italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , ⋅ , italic_w start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) and obtain an w2kDr2R2superscriptsubscript𝑤2𝑘subscript𝐷subscript𝑟2subscript𝑅2\ w_{2}^{k}\in D_{r_{2}R_{2}}italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∈ italic_D start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT with

J2(w1k,w2k,..,wnk1)infDr2R2J2(w1k,,w3k1,..,wnk1)+1k,J22(w1k,w2k,..,wnk1)H01(Ω)1k.J_{2}\left(w_{1}^{k},w_{2}^{k},..,w_{n}^{k-1}\right)\leq\inf_{D_{r_{2}R_{2}}}J% _{2}\left(w_{1}^{k},\cdot,w_{3}^{k-1},..,w_{n}^{k-1}\right)+\frac{1}{k},\ \ % \left\|J_{22}\left(w_{1}^{k},w_{2}^{k},..,w_{n}^{k-1}\right)\right\|_{H_{0}^{1% }\left(\Omega\right)}\leq\frac{1}{k}.italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) ≤ roman_inf start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , ⋅ , italic_w start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) + divide start_ARG 1 end_ARG start_ARG italic_k end_ARG , ∥ italic_J start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG italic_k end_ARG .

After n𝑛nitalic_n steps we obtain w1k,..,wnk,\ w_{1}^{k},..,w_{n}^{k},italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , where for each i{1,..,n},i\in\left\{1,..,n\right\},italic_i ∈ { 1 , . . , italic_n } , one has

Ji(w1k,..,wik,wi+1k1,..,wnk1)infDriRiJi(w1k,..,wi1k,,wi+1k1,..,wnk1)+1k,J_{i}\left(w_{1}^{k},..,w_{i}^{k},w_{i+1}^{k-1},..,w_{n}^{k-1}\right)\,\leq\,% \inf_{D_{r_{i}R_{i}}}J_{i}\left(w_{1}^{k},..,w_{i-1}^{k},\cdot,w_{i+1}^{k-1},.% .,w_{n}^{k-1}\right)+\frac{1}{k},italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) ≤ roman_inf start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , ⋅ , italic_w start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) + divide start_ARG 1 end_ARG start_ARG italic_k end_ARG ,
Jii(w1k,..,wik,wi+1k1,..,wnk1)H01(Ω)1k.\left\|J_{ii}\left(w_{1}^{k},..,w_{i}^{k},w_{i+1}^{k-1},..,w_{n}^{k-1}\right)% \right\|_{H_{0}^{1}\left(\Omega\right)}\,\leq\,\frac{1}{k}.∥ italic_J start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG italic_k end_ARG . (3.4)

Our next goal is to prove the convergence of the sequences (wik)k1(i=1,..,n).\left(w_{i}^{k}\right)_{k\geq 1}\ (i=1,..,n).\ ( italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k ≥ 1 end_POSTSUBSCRIPT ( italic_i = 1 , . . , italic_n ) .To this aim we use a technique first suggested in [24] and extended in [26]. For i=1,..,ni=1,..,nitalic_i = 1 , . . , italic_n and k1,𝑘1k\geq 1,italic_k ≥ 1 , we let

vik:=Jii(w1k,..,wik,wi+1k1,..,wnk1).v_{ik}:=J_{ii}\left(w_{1}^{k},..,w_{i}^{k},w_{i+1}^{k-1},..,w_{n}^{k-1}\right).italic_v start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT := italic_J start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) .

From (3.4), one has vik0subscript𝑣𝑖𝑘0\ v_{ik}\rightarrow 0italic_v start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT → 0 in H01(Ω)superscriptsubscript𝐻01ΩH_{0}^{1}\left(\Omega\right)italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) as k.𝑘k\rightarrow\infty.italic_k → ∞ . We have

wikNi(w1k,..,wik,wi+1k1,..,wnk1)=vikw_{i}^{k}-N_{i}\left(w_{1}^{k},..,w_{i}^{k},w_{i+1}^{k-1},..,w_{n}^{k-1}\right% )\ =v_{ik}italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT - italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) = italic_v start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT (3.5)

which yields

wik+pwikH01(Ω)subscriptnormsuperscriptsubscript𝑤𝑖𝑘𝑝superscriptsubscript𝑤𝑖𝑘superscriptsubscript𝐻01Ωabsent\displaystyle\left\|w_{i}^{k+p}-w_{i}^{k}\right\|_{H_{0}^{1}\left(\Omega\right% )}\leq∥ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_p end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ≤ (3.6)
Ni(w1k+p,..,wik+p,wi+1k+p1,..,wnk+p1)Ni(w1k,..,wik,wi+1k1,..,wnk1)H01(Ω)+vik+pvikH01(Ω)\displaystyle\left\|N_{i}\left(w_{1}^{k+p},..,w_{i}^{k+p},w_{i+1}^{k+p-1},..,w% _{n}^{k+p-1}\right)-N_{i}\left(w_{1}^{k},..,w_{i}^{k},w_{i+1}^{k-1},..,w_{n}^{% k-1}\right)\right\|_{H_{0}^{1}\left(\Omega\right)}+\left\|v_{ik+p}-v_{ik}% \right\|_{H_{0}^{1}\left(\Omega\right)}\leq∥ italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_p end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_p end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_p - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_p - 1 end_POSTSUPERSCRIPT ) - italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT , . . , italic_w start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT + ∥ italic_v start_POSTSUBSCRIPT italic_i italic_k + italic_p end_POSTSUBSCRIPT - italic_v start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ≤
j=1iLijwjk+pwjkH01(Ω)+j=i+1nLijwjk+p1wjk1H01(Ω)+vik+pvikH01(Ω).superscriptsubscript𝑗1𝑖subscript𝐿𝑖𝑗subscriptnormsuperscriptsubscript𝑤𝑗𝑘𝑝superscriptsubscript𝑤𝑗𝑘superscriptsubscript𝐻01Ωsuperscriptsubscript𝑗𝑖1𝑛subscript𝐿𝑖𝑗subscriptnormsuperscriptsubscript𝑤𝑗𝑘𝑝1superscriptsubscript𝑤𝑗𝑘1superscriptsubscript𝐻01Ωsubscriptnormsubscript𝑣𝑖𝑘𝑝subscript𝑣𝑖𝑘superscriptsubscript𝐻01Ω\displaystyle\sum\limits_{j=1}^{i}L_{ij}\left\|w_{j}^{k+p}-w_{j}^{k}\right\|_{% H_{0}^{1}\left(\Omega\right)}+\sum\limits_{j=i+1}^{n}L_{ij}\left\|w_{j}^{k+p-1% }-w_{j}^{k-1}\right\|_{H_{0}^{1}\left(\Omega\right)}+\left\|v_{ik+p}-v_{ik}% \right\|_{H_{0}^{1}\left(\Omega\right)}.∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_p end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_j = italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_p - 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - 1 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT + ∥ italic_v start_POSTSUBSCRIPT italic_i italic_k + italic_p end_POSTSUBSCRIPT - italic_v start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT .

Denote

ai,k,p=wik+pwikH01(Ω),bi,k,p=vik+pvikH01(Ω).formulae-sequencesubscript𝑎𝑖𝑘𝑝subscriptnormsuperscriptsubscript𝑤𝑖𝑘𝑝superscriptsubscript𝑤𝑖𝑘superscriptsubscript𝐻01Ωsubscript𝑏𝑖𝑘𝑝subscriptnormsubscript𝑣𝑖𝑘𝑝subscript𝑣𝑖𝑘superscriptsubscript𝐻01Ωa_{i,k,p}=\left\|w_{i}^{k+p}-w_{i}^{k}\right\|_{H_{0}^{1}\left(\Omega\right)},% \ \ b_{i,k,p}=\left\|v_{ik+p}-v_{ik}\right\|_{H_{0}^{1}\left(\Omega\right)}.italic_a start_POSTSUBSCRIPT italic_i , italic_k , italic_p end_POSTSUBSCRIPT = ∥ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_p end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_i , italic_k , italic_p end_POSTSUBSCRIPT = ∥ italic_v start_POSTSUBSCRIPT italic_i italic_k + italic_p end_POSTSUBSCRIPT - italic_v start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT .

One clearly has,

bi,k,p0 as k, uniformly with respect to p.formulae-sequencesubscript𝑏𝑖𝑘𝑝0 as 𝑘 uniformly with respect to 𝑝b_{i,k,p}\rightarrow 0\,\,\text{ as }k\rightarrow\infty,\,\text{ uniformly with respect to }p.italic_b start_POSTSUBSCRIPT italic_i , italic_k , italic_p end_POSTSUBSCRIPT → 0 as italic_k → ∞ , uniformly with respect to italic_p . (3.7)

With the notations before, inequality (3.6) writes

(a1,k,pa2,k,p..an,k,p)(L110..0L21L22..0........Ln1Ln2..Lnn)(a1,k,pa2,k,p..an,k,p)+(0L12..L1n00..L2n........0....0)(b1,k,pb2,k,p..bn,k,p).subscript𝑎1𝑘𝑝subscript𝑎2𝑘𝑝absentsubscript𝑎𝑛𝑘𝑝subscript𝐿110absent0subscript𝐿21subscript𝐿22absent0absentabsentabsentabsentsubscript𝐿𝑛1subscript𝐿𝑛2absentsubscript𝐿𝑛𝑛subscript𝑎1𝑘𝑝subscript𝑎2𝑘𝑝absentsubscript𝑎𝑛𝑘𝑝0subscript𝐿12absentsubscript𝐿1𝑛00absentsubscript𝐿2𝑛absentabsentabsentabsent0absentabsent0subscript𝑏1𝑘𝑝subscript𝑏2𝑘𝑝absentsubscript𝑏𝑛𝑘𝑝\left(\begin{array}[]{c}a_{1,k,p}\\ a_{2,k,p}\\ ..\\ a_{n,k,p}\end{array}\right)\leq\left(\begin{array}[]{cccc}L_{11}&0&..&0\\ L_{21}&L_{22}&..&0\\ ..&..&..&..\\ L_{n1}&L_{n2}&..&L_{nn}\end{array}\right)\left(\begin{array}[]{c}a_{1,k,p}\\ a_{2,k,p}\\ ..\\ a_{n,k,p}\end{array}\right)+\left(\begin{array}[]{cccc}0&L_{12}&..&L_{1n}\\ 0&0&..&L_{2n}\\ ..&..&..&..\\ 0&..&..&0\end{array}\right)\left(\begin{array}[]{c}b_{1,k,p}\\ b_{2,k,p}\\ ..\\ b_{n,k,p}\end{array}\right).( start_ARRAY start_ROW start_CELL italic_a start_POSTSUBSCRIPT 1 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT 2 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT italic_n , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) ≤ ( start_ARRAY start_ROW start_CELL italic_L start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL . . end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_L start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT end_CELL start_CELL italic_L start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL start_CELL . . end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL . . end_CELL start_CELL . . end_CELL start_CELL . . end_CELL start_CELL . . end_CELL end_ROW start_ROW start_CELL italic_L start_POSTSUBSCRIPT italic_n 1 end_POSTSUBSCRIPT end_CELL start_CELL italic_L start_POSTSUBSCRIPT italic_n 2 end_POSTSUBSCRIPT end_CELL start_CELL . . end_CELL start_CELL italic_L start_POSTSUBSCRIPT italic_n italic_n end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) ( start_ARRAY start_ROW start_CELL italic_a start_POSTSUBSCRIPT 1 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT 2 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT italic_n , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) + ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL italic_L start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT end_CELL start_CELL . . end_CELL start_CELL italic_L start_POSTSUBSCRIPT 1 italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL . . end_CELL start_CELL italic_L start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL start_CELL . . end_CELL start_CELL . . end_CELL start_CELL . . end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL . . end_CELL start_CELL . . end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY ) ( start_ARRAY start_ROW start_CELL italic_b start_POSTSUBSCRIPT 1 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_b start_POSTSUBSCRIPT 2 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL italic_b start_POSTSUBSCRIPT italic_n , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) .

By denoting =(aij)1i,jn,superscriptsubscriptsubscript𝑎𝑖𝑗formulae-sequence1𝑖𝑗𝑛\ \mathcal{M}^{\prime}=\left(a_{ij}\right)_{1\leq i,j\leq n},caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 1 ≤ italic_i , italic_j ≤ italic_n end_POSTSUBSCRIPT , where aij=Lijsubscript𝑎𝑖𝑗subscript𝐿𝑖𝑗\ a_{ij}=L_{ij}italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT for ji𝑗𝑖j\leq iitalic_j ≤ italic_i and aij=0subscript𝑎𝑖𝑗0\ a_{ij}=0italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = 0 for j>i,𝑗𝑖j>i,italic_j > italic_i , and ′′=,superscript′′superscript\ \mathcal{M}^{\prime\prime}=\mathcal{M}-\mathcal{M}^{\prime},caligraphic_M start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = caligraphic_M - caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , the above inequality can be written in the form

(a1,k,p..an,k,p)(a1,k,p..an,k,p)+′′(b1,k,p..bn,k,p).subscript𝑎1𝑘𝑝absentsubscript𝑎𝑛𝑘𝑝superscriptsubscript𝑎1𝑘𝑝absentsubscript𝑎𝑛𝑘𝑝superscript′′subscript𝑏1𝑘𝑝absentsubscript𝑏𝑛𝑘𝑝\left(\begin{array}[]{c}a_{1,k,p}\\ ..\\ a_{n,k,p}\end{array}\right)\leq\mathcal{M}^{\prime}\left(\begin{array}[]{c}a_{% 1,k,p}\\ ..\\ a_{n,k,p}\end{array}\right)+\mathcal{M}^{\prime\prime}\left(\begin{array}[]{c}% b_{1,k,p}\\ ..\\ b_{n,k,p}\end{array}\right).( start_ARRAY start_ROW start_CELL italic_a start_POSTSUBSCRIPT 1 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT italic_n , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) ≤ caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( start_ARRAY start_ROW start_CELL italic_a start_POSTSUBSCRIPT 1 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT italic_n , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) + caligraphic_M start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( start_ARRAY start_ROW start_CELL italic_b start_POSTSUBSCRIPT 1 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL italic_b start_POSTSUBSCRIPT italic_n , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) .

Since k0superscript𝑘0\ \mathcal{M}^{k}\rightarrow 0\ caligraphic_M start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT → 0as k,𝑘k\rightarrow\infty,italic_k → ∞ , we clearly have that ksuperscript𝑘\ \mathcal{M}^{\prime k}\rightarrow\inftycaligraphic_M start_POSTSUPERSCRIPT ′ italic_k end_POSTSUPERSCRIPT → ∞ as k,𝑘k\rightarrow\infty,italic_k → ∞ , so the matrix I𝐼superscript\ I-\mathcal{M}^{\prime}italic_I - caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is invertible and its inverse is nonnegative. Consequently

(a1,k,p..an,k,p)(I)1′′(b1,k,p..bn,k,p).subscript𝑎1𝑘𝑝absentsubscript𝑎𝑛𝑘𝑝superscript𝐼superscript1superscript′′subscript𝑏1𝑘𝑝absentsubscript𝑏𝑛𝑘𝑝\left(\begin{array}[]{c}a_{1,k,p}\\ ..\\ a_{n,k,p}\end{array}\right)\leq\left(I-\mathcal{M}^{\prime}\right)^{-1}% \mathcal{M}^{\prime\prime}\left(\begin{array}[]{c}b_{1,k,p}\\ ..\\ b_{n,k,p}\end{array}\right).( start_ARRAY start_ROW start_CELL italic_a start_POSTSUBSCRIPT 1 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL italic_a start_POSTSUBSCRIPT italic_n , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) ≤ ( italic_I - caligraphic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT caligraphic_M start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( start_ARRAY start_ROW start_CELL italic_b start_POSTSUBSCRIPT 1 , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . . end_CELL end_ROW start_ROW start_CELL italic_b start_POSTSUBSCRIPT italic_n , italic_k , italic_p end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) .

It follows that the sequences ai,k,psubscript𝑎𝑖𝑘𝑝a_{i,k,p}italic_a start_POSTSUBSCRIPT italic_i , italic_k , italic_p end_POSTSUBSCRIPT converge to zero as k𝑘k\rightarrow\inftyitalic_k → ∞ uniformly with respect to p.𝑝p.italic_p . Thus, for each i{1,..,n},i\in\left\{1,..,n\right\},italic_i ∈ { 1 , . . , italic_n } , the sequence (wik)ksubscriptsuperscriptsubscript𝑤𝑖𝑘𝑘\left(w_{i}^{k}\right)_{k}( italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is Cauchy in H01(Ω)superscriptsubscript𝐻01ΩH_{0}^{1}\left(\Omega\right)italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) and therefore convergent to some w¯i.subscript¯𝑤𝑖\overline{w}_{i}.over¯ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT . Passing to the limit in (3.4) we obtain

Jii(w¯)=0(i=1,..,n),J_{ii}\left(\overline{w}\right)=0\ \ \ \left(i=1,..,n\right),italic_J start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT ( over¯ start_ARG italic_w end_ARG ) = 0 ( italic_i = 1 , . . , italic_n ) , (3.8)
Ji(w¯)=infDriRiJi(w¯1,..,w¯i1,,w¯i+1,..,w¯n)(i=1,..,n).J_{i}\left(\overline{w}\right)=\inf_{D_{r_{i}R_{i}}}J_{i}\left(\overline{w}_{1% },..,\overline{w}_{i-1},\cdot,\overline{w}_{i+1},..,\overline{w}_{n}\right)\ % \ \ \left(i=1,..,n\right).italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over¯ start_ARG italic_w end_ARG ) = roman_inf start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over¯ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , . . , over¯ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT , ⋅ , over¯ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT , . . , over¯ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ( italic_i = 1 , . . , italic_n ) . (3.9)

Now (3.8) shows that w¯¯𝑤\ \overline{w}over¯ start_ARG italic_w end_ARG solves (2.9) in DrR,subscript𝐷𝑟𝑅D_{rR},italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT , and the uniqueness of solution implies that w¯=w,¯𝑤𝑤\ \overline{w}=w,over¯ start_ARG italic_w end_ARG = italic_w , where w𝑤\ witalic_w is the solution given by Perov’s theorem. Finally, (3.9) shows that w𝑤witalic_w is a Nash equilibrium with respect to the partial energy functionals (J1,..,Jn).\left(J_{1},..\ ,J_{n}\right).( italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , . . , italic_J start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) .   

In terms of the Stokes system, Theorem 3 immediately yields the following localization and multiplicity result.

Theorem 4

(a) Under conditions (h1) and (h2), problem (2.11) has a unique (modulo an additive constant in pressure) solution (u,p)𝑢𝑝\left(u,p\right)( italic_u , italic_p ) such that the recovered velocity u+q𝑢𝑞u+qitalic_u + italic_q belongs to DrRsubscript𝐷𝑟𝑅D_{rR}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT and is a Nash equilibrium with respect to the associated partial energy functionals.

(b) In case that conditions (h1) and (h2) are satisfied for two pairs of vectors (r,R)𝑟𝑅\left(r,R\right)( italic_r , italic_R ) and (r¯,R¯)¯𝑟¯𝑅\left(\overline{r},\overline{R}\right)( over¯ start_ARG italic_r end_ARG , over¯ start_ARG italic_R end_ARG ) with 0<r<Rr¯<R¯, 0𝑟𝑅less-than-and-not-equals¯𝑟¯𝑅\ 0<r<R\lvertneqq\overline{r}<\overline{R},0 < italic_r < italic_R ≨ over¯ start_ARG italic_r end_ARG < over¯ start_ARG italic_R end_ARG , problem (2.11) has two distinct solutions (u,p),𝑢𝑝\left(u,p\right),( italic_u , italic_p ) , (u¯,p¯)¯𝑢¯𝑝\left(\overline{u},\overline{p}\right)( over¯ start_ARG italic_u end_ARG , over¯ start_ARG italic_p end_ARG ) with u+qDrR𝑢𝑞subscript𝐷𝑟𝑅\ u+q\in D_{rR}italic_u + italic_q ∈ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT and u¯+q¯Dr¯R¯.¯𝑢¯𝑞subscript𝐷¯𝑟¯𝑅\ \overline{u}+\overline{q}\in D_{\overline{r}\overline{R}}.over¯ start_ARG italic_u end_ARG + over¯ start_ARG italic_q end_ARG ∈ italic_D start_POSTSUBSCRIPT over¯ start_ARG italic_r end_ARG over¯ start_ARG italic_R end_ARG end_POSTSUBSCRIPT .

3.2 Localization of the velocity


The results established in Section 3.1 clearly yield localization of the velocity u𝑢uitalic_u also in the particular case p=0,𝑝0\nabla p=0,∇ italic_p = 0 , that is for a uniform pressure. In this section we give an answer to the question concerning the type of forces which guarantee us to have a uniform pressure. More precisely, we give a sufficient condition for the external force f𝑓fitalic_f  in order to guarantee that p=0,𝑝0\nabla p=0,∇ italic_p = 0 , namely the equality

divΔL1f=0.divΔsuperscript𝐿1𝑓0\operatorname{div}\Delta L^{-1}f=0.roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f = 0 . (3.10)

We underline the dependence of the sufficient condition (3.10) on the viscosity μ(x)𝜇𝑥\mu\left(x\right)italic_μ ( italic_x ) by means of the operator L𝐿Litalic_L and the fact that in case of a constant viscosity this condition reduces to the self property of the force divf=0,div𝑓0\ \operatorname{div}f=0,roman_div italic_f = 0 , namely the force f𝑓fitalic_f is irrotational. For the same condition yielding constant pressure in the case of periodic boundary conditions, we refer the reader to [18, pag. 84].

Our aim is to investigate the localization of the velocity u𝑢uitalic_u for planar Stokes systems with an additional reaction force depending on the velocity itself, namely

{Lu+p=f(x)+ϕ[u](x)in Ωdivu=0in Ωu=0on Ω.cases𝐿𝑢𝑝𝑓𝑥italic-ϕdelimited-[]𝑢𝑥in Ωdiv𝑢0in Ω𝑢0on Ω\left\{\begin{array}[]{ll}Lu+\nabla p=f\left(x\right)+\phi\left[u\right]\left(% x\right)&\text{in }\Omega\\ \operatorname{div}\,u=0&\text{in }\Omega\\ u=0&\text{on }\partial\Omega.\end{array}\right.{ start_ARRAY start_ROW start_CELL italic_L italic_u + ∇ italic_p = italic_f ( italic_x ) + italic_ϕ [ italic_u ] ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u = 0 end_CELL start_CELL on ∂ roman_Ω . end_CELL end_ROW end_ARRAY (3.11)

Here fL2(Ω;2)𝑓superscript𝐿2Ωsuperscript2f\in L^{2}\left(\Omega;\mathbb{R}^{2}\right)italic_f ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) and ϕ:VL2(Ω;2).:italic-ϕ𝑉superscript𝐿2Ωsuperscript2\phi:V\rightarrow L^{2}\left(\Omega;\mathbb{R}^{2}\right).italic_ϕ : italic_V → italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) . We shall consider the special type of the reaction additional force, defined by

ϕ[u](x)=L(Δ)1(1Ωη(u(.)))(x),\phi\left[u\right]\left(x\right)=L\left(-\Delta\right)^{-1}\left(1_{\Omega^{% \prime}}\eta\left(u\left(.\right)\right)\right)\left(x\right),italic_ϕ [ italic_u ] ( italic_x ) = italic_L ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_η ( italic_u ( . ) ) ) ( italic_x ) , (3.12)

where η:22:𝜂superscript2superscript2\ \eta:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}italic_η : blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and the hypothesis on η𝜂\etaitalic_η will be made precise later.

We start with some preliminary results. We analyze first system (2.1) with a given non reactive external force f(x)𝑓𝑥f\left(x\right)italic_f ( italic_x ) satisfying the sufficient condition (3.10). One has the following result.

Lemma 5

If divΔL1f=0,divΔsuperscript𝐿1𝑓0\ \operatorname{div}\Delta L^{-1}f=0,roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f = 0 , then p=0𝑝0\ \nabla p=0∇ italic_p = 0 and Sf=L1f𝑆𝑓superscript𝐿1𝑓\ Sf=L^{-1}fitalic_S italic_f = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f, where S𝑆Sitalic_S is the velocity operator defined in (2.4).

Proof. The function u¯=L1fH01(Ω;n)¯𝑢superscript𝐿1𝑓superscriptsubscript𝐻01Ωsuperscript𝑛\ \overline{u}=L^{-1}f\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right)over¯ start_ARG italic_u end_ARG = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) satisfies

((u¯,v))H01(f,v)=0for all vH01(Ω;n).formulae-sequencesubscript¯𝑢𝑣superscriptsubscript𝐻01𝑓𝑣0for all 𝑣superscriptsubscript𝐻01Ωsuperscript𝑛\left(\left(\overline{u},v\right)\right)_{H_{0}^{1}}-\left(f,v\right)=0\ \ \ % \text{for all \ }v\in H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right).( ( over¯ start_ARG italic_u end_ARG , italic_v ) ) start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - ( italic_f , italic_v ) = 0 for all italic_v ∈ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) .

In particular, since VH01(Ω;n),𝑉superscriptsubscript𝐻01Ωsuperscript𝑛V\subset H_{0}^{1}\left(\Omega;\mathbb{R}^{n}\right),italic_V ⊂ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) , u¯¯𝑢\overline{u}over¯ start_ARG italic_u end_ARG satisfies the identity (2.3). It remains to prove that u¯V,¯𝑢𝑉\overline{u}\in V,over¯ start_ARG italic_u end_ARG ∈ italic_V , that is divu¯=0.div¯𝑢0\ \operatorname{div}\,\overline{u}=0.roman_div over¯ start_ARG italic_u end_ARG = 0 . It suffices to prove that the function divu¯div¯𝑢\ \operatorname{div}\,\overline{u}roman_div over¯ start_ARG italic_u end_ARG is zero on any subset ΩΩ.\Omega^{\prime}\subset\subset\Omega.\ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ ⊂ roman_Ω .Since C0(Ω)superscriptsubscript𝐶0superscriptΩC_{0}^{\infty}\left(\Omega^{\prime}\right)italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) is dense in L2(Ω),superscript𝐿2superscriptΩL^{2}\left(\Omega^{\prime}\right),italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , it is enough to show that

(divu¯,φ)L2(Ω)=0for all φC0(Ω).formulae-sequencesubscriptdiv¯𝑢𝜑superscript𝐿2superscriptΩ0for all 𝜑superscriptsubscript𝐶0superscriptΩ\left(\operatorname{div}\,\overline{u},\varphi\right)_{L^{2}\left(\Omega^{% \prime}\right)}=0\ \ \ \text{for all }\varphi\in C_{0}^{\infty}\left(\Omega^{% \prime}\right).( roman_div over¯ start_ARG italic_u end_ARG , italic_φ ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT = 0 for all italic_φ ∈ italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) . (3.13)

From the equality divΔL1f=0,divΔsuperscript𝐿1𝑓0\ \operatorname{div}\Delta L^{-1}f=0,roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f = 0 , one has, for any ψ𝜓absent\psi\initalic_ψ ∈ C0(Ω)superscriptsubscript𝐶0superscriptΩC_{0}^{\infty}\left(\Omega^{\prime}\right)italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )

00\displaystyle 0 =\displaystyle== (divΔL1f,ψ)=i=1n(ΔL1fixi,ψ)divΔsuperscript𝐿1𝑓𝜓superscriptsubscript𝑖1𝑛Δsuperscript𝐿1subscript𝑓𝑖subscript𝑥𝑖𝜓\displaystyle\left(\operatorname{div}\Delta L^{-1}f,\psi\right)=\sum\limits_{i% =1}^{n}\left(\frac{\partial\Delta L^{-1}f_{i}}{\partial x_{i}},\psi\right)( roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f , italic_ψ ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( divide start_ARG ∂ roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG , italic_ψ )
=\displaystyle== i=1n(ΔL1fixi,ψ)=i=1n(L1fixi,Δψ)L2(Ω)superscriptsubscript𝑖1𝑛Δsuperscript𝐿1subscript𝑓𝑖subscript𝑥𝑖𝜓superscriptsubscript𝑖1𝑛subscriptsuperscript𝐿1subscript𝑓𝑖subscript𝑥𝑖Δ𝜓superscript𝐿2superscriptΩ\displaystyle\sum\limits_{i=1}^{n}\left(\Delta\frac{\partial L^{-1}f_{i}}{% \partial x_{i}},\psi\right)=\sum\limits_{i=1}^{n}\left(\frac{\partial L^{-1}f_% {i}}{\partial x_{i}},\Delta\psi\right)_{L^{2}\left(\Omega^{\prime}\right)}∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( roman_Δ divide start_ARG ∂ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG , italic_ψ ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( divide start_ARG ∂ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG , roman_Δ italic_ψ ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT
=\displaystyle== (divu¯,Δψ)L2(Ω).subscriptdiv¯𝑢Δ𝜓superscript𝐿2superscriptΩ\displaystyle\left(\operatorname{div}\,\overline{u},\Delta\psi\right)_{L^{2}% \left(\Omega^{\prime}\right)}.( roman_div over¯ start_ARG italic_u end_ARG , roman_Δ italic_ψ ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT .

The symbol (.,.)\left(.,.\right)( . , . ) is used to denote the value of a distribution on a test function. Thus

(divu¯,Δψ)L2(Ω)=0for all ψC0(Ω).formulae-sequencesubscriptdiv¯𝑢Δ𝜓superscript𝐿2superscriptΩ0for all 𝜓superscriptsubscript𝐶0superscriptΩ\left(\operatorname{div}\,\overline{u},\Delta\psi\right)_{L^{2}\left(\Omega^{% \prime}\right)}=0\ \ \ \text{for all }\psi\in C_{0}^{\infty}\left(\Omega^{% \prime}\right).( roman_div over¯ start_ARG italic_u end_ARG , roman_Δ italic_ψ ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT = 0 for all italic_ψ ∈ italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) .

Now let φC0(Ω)𝜑superscriptsubscript𝐶0superscriptΩ\varphi\in C_{0}^{\infty}\left(\Omega^{\prime}\right)italic_φ ∈ italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) be arbitrary. Denote ψ:=(Δ)1φ.assign𝜓superscriptΔ1𝜑\psi:=\left(-\Delta\right)^{-1}\varphi.italic_ψ := ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_φ . Then Δψ=φ.Δ𝜓𝜑-\Delta\psi=\varphi.- roman_Δ italic_ψ = italic_φ . Consider a sequence ψkC0(Ω)subscript𝜓𝑘superscriptsubscript𝐶0superscriptΩ\ \psi_{k}\in C_{0}^{\infty}\left(\Omega^{\prime}\right)italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ italic_C start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) with ψkψsubscript𝜓𝑘𝜓\ \psi_{k}\rightarrow\psiitalic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT → italic_ψ in H01(Ω).superscriptsubscript𝐻01superscriptΩH_{0}^{1}\left(\Omega^{\prime}\right).italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) . Since μ𝜇\ \muitalic_μ is uniformly Lipschitz continuous and fL2(Ω;n),𝑓superscript𝐿2Ωsuperscript𝑛f\in L^{2}\left(\Omega;\mathbb{R}^{n}\right),italic_f ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) , one has u¯H2(Ω;n)¯𝑢superscript𝐻2superscriptΩsuperscript𝑛\ \overline{u}\in H^{2}\left(\Omega^{\prime};\mathbb{R}^{n}\right)over¯ start_ARG italic_u end_ARG ∈ italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) (see [13, Theorem 8.8]). Consequently

xidivu¯L2(Ω),i=1,..,n.\frac{\partial}{\partial x_{i}}\operatorname{div}\,\overline{u}\in L^{2}\left(% \Omega^{\prime}\right),\ \ \ i=1,..,n.divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG roman_div over¯ start_ARG italic_u end_ARG ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , italic_i = 1 , . . , italic_n .

We have

0=(divu¯,Δψk)L2(Ω)=i=1n(xidivu¯,ψkxi)L2(Ω).0subscriptdiv¯𝑢Δsubscript𝜓𝑘superscript𝐿2superscriptΩsuperscriptsubscript𝑖1𝑛subscriptsubscript𝑥𝑖div¯𝑢subscript𝜓𝑘subscript𝑥𝑖superscript𝐿2superscriptΩ0=\left(\operatorname{div}\,\overline{u},\Delta\psi_{k}\right)_{L^{2}\left(% \Omega^{\prime}\right)}=-\sum\limits_{i=1}^{n}\left(\frac{\partial}{\partial x% _{i}}\operatorname{div}\,\overline{u},\frac{\partial\psi_{k}}{\partial x_{i}}% \right)_{L^{2}\left(\Omega^{\prime}\right)}.0 = ( roman_div over¯ start_ARG italic_u end_ARG , roman_Δ italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT = - ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG roman_div over¯ start_ARG italic_u end_ARG , divide start_ARG ∂ italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT .

From ψkψsubscript𝜓𝑘𝜓\psi_{k}\rightarrow\psiitalic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT → italic_ψ in H01(Ω)superscriptsubscript𝐻01superscriptΩH_{0}^{1}\left(\Omega^{\prime}\right)italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) one has ψkxiψxi(i=1,..,n)\displaystyle\frac{\partial\psi_{k}}{\partial x_{i}}\rightarrow\frac{\partial% \psi}{\partial x_{i}}\ \left(i=1,..,n\right)divide start_ARG ∂ italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG → divide start_ARG ∂ italic_ψ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ( italic_i = 1 , . . , italic_n ) in L2(Ω).superscript𝐿2superscriptΩL^{2}\left(\Omega^{\prime}\right).italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) . Passing to the limit gives

00\displaystyle 0 =\displaystyle== i=1n(xidivu¯,ψxi)L2(Ω)=i=1n(divu¯,2ψxi2)L2(Ω)superscriptsubscript𝑖1𝑛subscriptsubscript𝑥𝑖𝑑𝑖𝑣¯𝑢𝜓subscript𝑥𝑖superscript𝐿2superscriptΩsuperscriptsubscript𝑖1𝑛subscriptdiv¯𝑢superscript2𝜓superscriptsubscript𝑥𝑖2superscript𝐿2superscriptΩ\displaystyle\sum\limits_{i=1}^{n}\left(\frac{\partial}{\partial x_{i}}% \operatorname{\,}{div}\,\overline{u},\frac{\partial\psi}{\partial x_{i}}\right% )_{L^{2}\left(\Omega^{\prime}\right)}=-\sum\limits_{i=1}^{n}\left(% \operatorname{div}\,\overline{u},\frac{\partial^{2}\psi}{\partial x_{i}^{2}}% \right)_{L^{2}\left(\Omega^{\prime}\right)}∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG italic_d italic_i italic_v over¯ start_ARG italic_u end_ARG , divide start_ARG ∂ italic_ψ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT = - ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( roman_div over¯ start_ARG italic_u end_ARG , divide start_ARG ∂ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ψ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT
=\displaystyle== (divu¯,Δψ)L2(Ω)=(divu¯,φ)L2(Ω).subscriptdiv¯𝑢Δ𝜓superscript𝐿2superscriptΩsubscriptdiv¯𝑢𝜑superscript𝐿2superscriptΩ\displaystyle-\left(\operatorname{div}\,\overline{u},\Delta\psi\right)_{L^{2}% \left(\Omega^{\prime}\right)}=\left(\operatorname{div}\,\overline{u},\varphi% \right)_{L^{2}\left(\Omega^{\prime}\right)}.- ( roman_div over¯ start_ARG italic_u end_ARG , roman_Δ italic_ψ ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT = ( roman_div over¯ start_ARG italic_u end_ARG , italic_φ ) start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT .

Thus (3.13) is proved.   

Remark 6

If f𝑓fitalic_f is such that divΔL1f=0,divΔsuperscript𝐿1𝑓0\ \operatorname{div}\Delta L^{-1}f=0,roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f = 0 , then denoting h:=ΔL1fassignΔsuperscript𝐿1𝑓\ h:=-\Delta L^{-1}fitalic_h := - roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f  one has the following expression of f,𝑓f,italic_f ,

f=L(Δ)1h=div(μ(x)(Δ)1h),f=L\left(-\Delta\right)^{-1}h=-\operatorname{div}\left(\mu\left(x\right)\nabla% \left(-\Delta\right)^{-1}h\right),italic_f = italic_L ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h = - roman_div ( italic_μ ( italic_x ) ∇ ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h ) ,

where divh=0.div0\ \operatorname{div}h=0.roman_div italic_h = 0 . This formula shows that for an external force to make the pressure constant it has to depend on the viscosity.

Note that since ΩΩ\Omegaroman_Ω is C2,superscript𝐶2C^{2},\ italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,if hL2(Ω;n)superscript𝐿2Ωsuperscript𝑛h\in L^{2}\left(\Omega;\mathbb{R}^{n}\right)italic_h ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) then (Δ)1hH2(Ω;n)superscriptΔ1superscript𝐻2Ωsuperscript𝑛\left(-\Delta\right)^{-1}h\in H^{2}\left(\Omega;\mathbb{R}^{n}\right)( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h ∈ italic_H start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) (see [13, Theorem 8.12]), which together with μC1(Ω¯)𝜇superscript𝐶1¯Ω\mu\in C^{1}\left(\overline{\Omega}\right)italic_μ ∈ italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( over¯ start_ARG roman_Ω end_ARG ) gives fL2(Ω;n).𝑓superscript𝐿2Ωsuperscript𝑛f\in L^{2}\left(\Omega;\mathbb{R}^{n}\right).italic_f ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) .

We show now that a localization of h\ hitalic_h immediately implies the localization of u.𝑢u.italic_u .

Proposition 7

Assume that divΔL1f=0.divΔsuperscript𝐿1𝑓0\ \operatorname{div}\Delta L^{-1}f=0.\ roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f = 0 .Let ΩΩ\ \Omega^{\prime}\subset\subset\Omegaroman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ ⊂ roman_Ω and r,R(0,)n𝑟𝑅superscript0𝑛\ r,R\in\left(0,\infty\right)^{n}italic_r , italic_R ∈ ( 0 , ∞ ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be such that 0<rγ<RΓ, 0𝑟𝛾𝑅Γ\ \ 0<\frac{r}{\gamma}<\frac{R}{\Gamma},0 < divide start_ARG italic_r end_ARG start_ARG italic_γ end_ARG < divide start_ARG italic_R end_ARG start_ARG roman_Γ end_ARG , where Γ=(Δ)11ΓsubscriptnormsuperscriptΔ11\Gamma=\left\|\left(-\Delta\right)^{-1}1\right\|_{\infty}roman_Γ = ∥ ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT 1 ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT and γ𝛾\gammaitalic_γ is such that (Δ)11Ω(x)γsuperscriptΔ1subscript1superscriptΩ𝑥𝛾\left(-\Delta\right)^{-1}1_{\Omega^{\prime}}\left(x\right)\geq\gamma( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) ≥ italic_γ on Ω.superscriptΩ\Omega^{\prime}.roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT . If

0(ΔL1f)(x)RΓ for a.e. xΩ and0Δsuperscript𝐿1𝑓𝑥𝑅Γ for a.e. 𝑥Ω and0\leq\left(-\Delta L^{-1}f\right)\left(x\right)\leq\frac{R}{\Gamma}\text{\ \ % \ for a.e. }x\in\Omega\ \text{ and}0 ≤ ( - roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ) ( italic_x ) ≤ divide start_ARG italic_R end_ARG start_ARG roman_Γ end_ARG for a.e. italic_x ∈ roman_Ω and (3.14)
rγ(ΔL1f)(x) for a.e. xΩ,𝑟𝛾Δsuperscript𝐿1𝑓𝑥 for a.e. 𝑥superscriptΩ\frac{r}{\gamma}\leq\left(-\Delta L^{-1}f\right)\left(x\right)\text{\ \ \ for % a.e. }x\in\Omega^{\prime},divide start_ARG italic_r end_ARG start_ARG italic_γ end_ARG ≤ ( - roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ) ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , (3.15)

then the velocity u𝑢uitalic_u satisfies

00\displaystyle 0 \displaystyle\leq u(x)R for a.e. xΩ andformulae-sequence𝑢𝑥𝑅 for a.e. 𝑥Ω and\displaystyle u\left(x\right)\leq R\ \ \ \text{\ for a.e. }x\in\Omega\text{\ % \ and}italic_u ( italic_x ) ≤ italic_R for a.e. italic_x ∈ roman_Ω and
r𝑟\displaystyle ritalic_r \displaystyle\leq u(x) for a.e. xΩ.𝑢𝑥 for a.e. 𝑥superscriptΩ\displaystyle u\left(x\right)\text{\ \ \ for a.e. }x\in\Omega^{\prime}.italic_u ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

Proof. From Lemma 5, one has u=Sf=L1f.𝑢𝑆𝑓superscript𝐿1𝑓\ u=Sf=L^{-1}f.\ italic_u = italic_S italic_f = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f .Next, from (3.14) and the remark (R2), we have

0((Δ)1h)(x)=u(x)RΓ((Δ)11)(x)Rfor a.e. xΩ.formulae-sequence0superscriptΔ1𝑥𝑢𝑥𝑅ΓsuperscriptΔ11𝑥𝑅for a.e. 𝑥Ω0\leq\left(\left(-\Delta\right)^{-1}h\right)\left(x\right)=u\left(x\right)\leq% \frac{R}{\Gamma}\left(\left(-\Delta\right)^{-1}1\right)\left(x\right)\leq R\ % \ \ \text{for a.e. }x\in\Omega.0 ≤ ( ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_h ) ( italic_x ) = italic_u ( italic_x ) ≤ divide start_ARG italic_R end_ARG start_ARG roman_Γ end_ARG ( ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT 1 ) ( italic_x ) ≤ italic_R for a.e. italic_x ∈ roman_Ω .

From   (3.15)   and 0(ΔL1f)(x) 0Δsuperscript𝐿1𝑓𝑥\ 0\leq\left(-\Delta L^{-1}f\right)\left(x\right)0 ≤ ( - roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ) ( italic_x )  in Ω,Ω\Omega,\ roman_Ω ,we have (ΔL1f)(x)rγ1Ω(x)Δsuperscript𝐿1𝑓𝑥𝑟𝛾subscript1superscriptΩ𝑥\displaystyle\ \left(-\Delta L^{-1}f\right)\left(x\right)\geq\frac{r}{\gamma}1% _{\Omega^{\prime}}\left(x\right)\ ( - roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ) ( italic_x ) ≥ divide start_ARG italic_r end_ARG start_ARG italic_γ end_ARG 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) for a.e. xΩ,𝑥Ωx\in\Omega,italic_x ∈ roman_Ω , whence

u(x)𝑢𝑥\displaystyle u\left(x\right)italic_u ( italic_x ) =\displaystyle== (Δ)1(ΔL1f)(x)(Δ)1(rγ1Ω)(x)superscriptΔ1Δsuperscript𝐿1𝑓𝑥superscriptΔ1𝑟𝛾subscript1superscriptΩ𝑥\displaystyle\left(-\Delta\right)^{-1}\left(-\Delta L^{-1}f\right)\left(x% \right)\geq\left(-\Delta\right)^{-1}\left(\frac{r}{\gamma}1_{\Omega^{\prime}}% \right)\left(x\right)( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( - roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ) ( italic_x ) ≥ ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( divide start_ARG italic_r end_ARG start_ARG italic_γ end_ARG 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ( italic_x )
=\displaystyle== rγ((Δ)11Ω)(x)r for a.e. xΩ.formulae-sequence𝑟𝛾superscriptΔ1subscript1superscriptΩ𝑥𝑟 for a.e. 𝑥superscriptΩ\displaystyle\frac{r}{\gamma}\left(\left(-\Delta\right)^{-1}1_{\Omega^{\prime}% }\right)\left(x\right)\geq r\ \ \text{\ for a.e. }x\in\Omega^{\prime}.divide start_ARG italic_r end_ARG start_ARG italic_γ end_ARG ( ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ( italic_x ) ≥ italic_r for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

 

Let us mention that in the particular case of a constant viscosity μ+,𝜇subscriptsuperscript\mu\in{\mathbb{R}}^{*}_{+},italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , one has L=μΔ𝐿𝜇Δ\ L=-\mu\Deltaitalic_L = - italic_μ roman_Δ and assumptions (3.14), (3.15) read as

0f(x)RμΓ for a.e. xΩ,rμγf(x) for a.e. xΩ.formulae-sequence0𝑓𝑥𝑅𝜇Γ for a.e. 𝑥Ω𝑟𝜇𝛾𝑓𝑥 for a.e. 𝑥superscriptΩ0\leq f\left(x\right)\leq\frac{R\mu}{\Gamma}\text{\ \ \ for a.e. }x\in\Omega,% \ \ \ \ \frac{r\mu}{\gamma}\leq f\left(x\right)\text{\ \ \ for a.e. }x\in\Omega^{\prime}.0 ≤ italic_f ( italic_x ) ≤ divide start_ARG italic_R italic_μ end_ARG start_ARG roman_Γ end_ARG for a.e. italic_x ∈ roman_Ω , divide start_ARG italic_r italic_μ end_ARG start_ARG italic_γ end_ARG ≤ italic_f ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

Next we consider planar Stokes systems with a velocity-dependent reaction force. It is convenient that besides the property divu=0,div𝑢0\operatorname{div}\,u=0,roman_div italic_u = 0 , to have divus=0,divsuperscript𝑢𝑠0\operatorname{div}\,u^{s}=0,roman_div italic_u start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = 0 , where for a given vector v=(v1,v2),𝑣subscript𝑣1subscript𝑣2v=\left(v_{1},v_{2}\right),italic_v = ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , vssuperscript𝑣𝑠v^{s}italic_v start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT is the vector defined by vs=(v2,v1).superscript𝑣𝑠subscript𝑣2subscript𝑣1v^{s}=\left(v_{2},v_{1}\right).italic_v start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = ( italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) . We have the two propositions below.

Proposition 8

If fL2(Ω;2)𝑓superscript𝐿2Ωsuperscript2\ f\in L^{2}\left(\Omega;\mathbb{R}^{2}\right)italic_f ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) is such that

divΔL1f=divΔL1fs=0,divΔsuperscript𝐿1𝑓divΔsuperscript𝐿1superscript𝑓𝑠0\operatorname{div}\Delta L^{-1}f=\operatorname{div}\Delta L^{-1}f^{s}=0,roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f = roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = 0 ,

then (Sf)sV.superscript𝑆𝑓𝑠𝑉\ \left(Sf\right)^{s}\in V.( italic_S italic_f ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ∈ italic_V .

Proof. According to Lemma 5, Sfs=L1fs.𝑆superscript𝑓𝑠superscript𝐿1superscript𝑓𝑠\ Sf^{s}=L^{-1}f^{s}.italic_S italic_f start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT . Clearly, one has L1fs=(L1f)s.superscript𝐿1superscript𝑓𝑠superscriptsuperscript𝐿1𝑓𝑠\ L^{-1}f^{s}=\left(L^{-1}f\right)^{s}.italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = ( italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT . Hence (Sf)s=SfsV.superscript𝑆𝑓𝑠𝑆superscript𝑓𝑠𝑉\ \left(Sf\right)^{s}=Sf^{s}\in V.( italic_S italic_f ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = italic_S italic_f start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ∈ italic_V .   

Let us now consider a closed subspace of V,𝑉V,italic_V , namely

Vs:={uV:usV}.assignsuperscript𝑉𝑠conditional-set𝑢𝑉superscript𝑢𝑠𝑉V^{s}:=\left\{u\in V:\ u^{s}\in V\right\}.italic_V start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT := { italic_u ∈ italic_V : italic_u start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ∈ italic_V } .

Also denote

X={fL2(Ω;2):divΔL1f=0},Xs={fX:divΔL1fs=0}.formulae-sequence𝑋conditional-set𝑓superscript𝐿2Ωsuperscript2divΔsuperscript𝐿1𝑓0superscript𝑋𝑠conditional-set𝑓𝑋divΔsuperscript𝐿1superscript𝑓𝑠0X=\left\{f\in L^{2}\left(\Omega;\mathbb{R}^{2}\right):\ \operatorname{div}% \Delta L^{-1}f=0\right\},\ \ \ \ X^{s}=\left\{f\in X:\ \operatorname{div}% \Delta L^{-1}f^{s}=0\right\}.italic_X = { italic_f ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) : roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f = 0 } , italic_X start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = { italic_f ∈ italic_X : roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = 0 } .

Hence one has

S(Xs)Vs,Sf=L1f(fX).formulae-sequence𝑆superscript𝑋𝑠superscript𝑉𝑠𝑆𝑓superscript𝐿1𝑓𝑓𝑋S\left(X^{s}\right)\subset V^{s},\ \ \ \ \ \ Sf=L^{-1}f\ \ \ \left(f\in X% \right).italic_S ( italic_X start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ) ⊂ italic_V start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT , italic_S italic_f = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ( italic_f ∈ italic_X ) .

We are now in position to consider the planar Stokes system (3.11) with the special type of reaction additional force defined by (3.12). We assume that η::𝜂absent\ \eta:italic_η : 22superscript2superscript2\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is of class C1superscript𝐶1C^{1}italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and

η1u1=η2u2,η1u2=η2u1.formulae-sequencesubscript𝜂1subscript𝑢1subscript𝜂2subscript𝑢2subscript𝜂1subscript𝑢2subscript𝜂2subscript𝑢1\frac{\partial\eta_{1}}{\partial u_{1}}=\frac{\partial\eta_{2}}{\partial u_{2}% },\ \ \ \ \frac{\partial\eta_{1}}{\partial u_{2}}=\frac{\partial\eta_{2}}{% \partial u_{1}}.divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG = divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG , divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG = divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG . (3.16)

One has the following result.

Proposition 9

Under assumption (3.16), η(u)𝜂𝑢\eta\left(u\right)italic_η ( italic_u ) belongs to Xssuperscript𝑋𝑠X^{s}italic_X start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT for every uVs.𝑢superscript𝑉𝑠u\in V^{s}.italic_u ∈ italic_V start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT .

Proof. We have to show that divη(u)=divη(u)s=0div𝜂𝑢div𝜂superscript𝑢𝑠0\ \operatorname{div}\,\eta\left(u\right)=\operatorname{div}\,\eta\left(u\right% )^{s}=0roman_div italic_η ( italic_u ) = roman_div italic_η ( italic_u ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = 0 in Ω.Ω\Omega.roman_Ω . Indeed, using (3.16) and divu=divus=0div𝑢divsuperscript𝑢𝑠0\operatorname{div}\,u=\operatorname{div}\,u^{s}=0roman_div italic_u = roman_div italic_u start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = 0 we deduce

divη(u)div𝜂𝑢\displaystyle\operatorname{div}\,\eta\left(u\right)roman_div italic_η ( italic_u ) =\displaystyle== x1η1(u)+x2η2(u)subscript𝑥1subscript𝜂1𝑢subscript𝑥2subscript𝜂2𝑢\displaystyle\frac{\partial}{\partial x_{1}}\eta_{1}\left(u\right)+\frac{% \partial}{\partial x_{2}}\eta_{2}\left(u\right)divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u ) + divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_u )
=\displaystyle== η1(u)u1u1x1+η1(u)u2u2x1+η2(u)u1u1x2+η2(u)u2u2x2subscript𝜂1𝑢subscript𝑢1subscript𝑢1subscript𝑥1subscript𝜂1𝑢subscript𝑢2subscript𝑢2subscript𝑥1subscript𝜂2𝑢subscript𝑢1subscript𝑢1subscript𝑥2subscript𝜂2𝑢subscript𝑢2subscript𝑢2subscript𝑥2\displaystyle\frac{\partial\eta_{1}\left(u\right)}{\partial u_{1}}\frac{% \partial u_{1}}{\partial x_{1}}+\frac{\partial\eta_{1}\left(u\right)}{\partial u% _{2}}\frac{\partial u_{2}}{\partial x_{1}}+\frac{\partial\eta_{2}\left(u\right% )}{\partial u_{1}}\frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial\eta_{2}% \left(u\right)}{\partial u_{2}}\frac{\partial u_{2}}{\partial x_{2}}divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG
=\displaystyle== η1(u)u1divu+η1(u)u2divus=0.subscript𝜂1𝑢subscript𝑢1div𝑢subscript𝜂1𝑢subscript𝑢2divsuperscript𝑢𝑠0\displaystyle\frac{\partial\eta_{1}\left(u\right)}{\partial u_{1}}% \operatorname{div}\,u+\frac{\partial\eta_{1}\left(u\right)}{\partial u_{2}}% \operatorname{div}\,u^{s}=0.divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_div italic_u + divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG roman_div italic_u start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = 0 .

Similarly one has

divη(u)sdiv𝜂superscript𝑢𝑠\displaystyle\operatorname{div}\,\eta\left(u\right)^{s}roman_div italic_η ( italic_u ) start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT =\displaystyle== x1η2(u)+x2η1(u)subscript𝑥1subscript𝜂2𝑢subscript𝑥2subscript𝜂1𝑢\displaystyle\frac{\partial}{\partial x_{1}}\eta_{2}\left(u\right)+\frac{% \partial}{\partial x_{2}}\eta_{1}\left(u\right)divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_u ) + divide start_ARG ∂ end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u )
=\displaystyle== η2(u)u1u1x1+η2(u)u2u2x1+η1(u)u1u1x2+η1(u)u2u2x2subscript𝜂2𝑢subscript𝑢1subscript𝑢1subscript𝑥1subscript𝜂2𝑢subscript𝑢2subscript𝑢2subscript𝑥1subscript𝜂1𝑢subscript𝑢1subscript𝑢1subscript𝑥2subscript𝜂1𝑢subscript𝑢2subscript𝑢2subscript𝑥2\displaystyle\frac{\partial\eta_{2}\left(u\right)}{\partial u_{1}}\frac{% \partial u_{1}}{\partial x_{1}}+\frac{\partial\eta_{2}\left(u\right)}{\partial u% _{2}}\frac{\partial u_{2}}{\partial x_{1}}+\frac{\partial\eta_{1}\left(u\right% )}{\partial u_{1}}\frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial\eta_{1}% \left(u\right)}{\partial u_{2}}\frac{\partial u_{2}}{\partial x_{2}}divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG + divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG divide start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG
=\displaystyle== η1(u)u1divus+η1(u)u2divu=0.subscript𝜂1𝑢subscript𝑢1divsuperscript𝑢𝑠subscript𝜂1𝑢subscript𝑢2div𝑢0\displaystyle\frac{\partial\eta_{1}\left(u\right)}{\partial u_{1}}% \operatorname{div}\,u^{s}+\frac{\partial\eta_{1}\left(u\right)}{\partial u_{2}% }\operatorname{div}\,u=0.divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG roman_div italic_u start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT + divide start_ARG ∂ italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_u ) end_ARG start_ARG ∂ italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG roman_div italic_u = 0 .

 

Assume that fXs,𝑓superscript𝑋𝑠\ f\in X^{s},italic_f ∈ italic_X start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT , i.e., fL2(Ω;2)𝑓superscript𝐿2Ωsuperscript2f\in L^{2}\left(\Omega;\mathbb{R}^{2}\right)italic_f ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) and divΔL1f=divΔL1fs=0.divΔsuperscript𝐿1𝑓divΔsuperscript𝐿1superscript𝑓𝑠0\ \operatorname{div}\Delta L^{-1}f=\operatorname{div}\Delta L^{-1}f^{s}=0.\ roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f = roman_div roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = 0 . We give the fixed point formulation of problem (3.11) (under hypothesis (3.12)). Under the above conditions, we have

η:VsXs,S:XsVs.:𝜂superscript𝑉𝑠superscript𝑋𝑠𝑆:superscript𝑋𝑠superscript𝑉𝑠\eta:V^{s}\rightarrow X^{s},\ \ \ S:X^{s}\rightarrow V^{s}.italic_η : italic_V start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT → italic_X start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT , italic_S : italic_X start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT → italic_V start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT .

Hence

T:=S(f+ϕ[u])=L1(f+ϕ[u])=(Δ)1(h+1Ωη(u)):VsVs:assign𝑇𝑆𝑓italic-ϕdelimited-[]𝑢superscript𝐿1𝑓italic-ϕdelimited-[]𝑢superscriptΔ1subscript1superscriptΩ𝜂𝑢superscript𝑉𝑠superscript𝑉𝑠T:=S\left(f+\phi\left[u\right]\right)=L^{-1}\left(f+\phi\left[u\right]\right)=% \left(-\Delta\right)^{-1}\left(h+1_{\Omega^{\prime}}\eta\left(u\right)\right):% V^{s}\rightarrow V^{s}italic_T := italic_S ( italic_f + italic_ϕ [ italic_u ] ) = italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_f + italic_ϕ [ italic_u ] ) = ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_h + 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_η ( italic_u ) ) : italic_V start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT → italic_V start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT

and the following result holds.

Proposition 10

(a) Problem (3.11) has a solution if and only if the operator T𝑇Titalic_T has a fixed point.

(b) If (u,p)𝑢𝑝\left(u,p\right)( italic_u , italic_p ) solves (3.11), then u𝑢uitalic_u is a fixed point of T𝑇Titalic_T and p=c𝑝𝑐p=citalic_p = italic_c (constant).

(c) If u𝑢uitalic_u is a fixed point of T,𝑇T,italic_T , then (u,c)𝑢𝑐\left(u,c\right)( italic_u , italic_c ) solves (3.11) for any constant c.𝑐c.italic_c .

Let DrRssuperscriptsubscript𝐷𝑟𝑅𝑠D_{rR}^{s}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT be defined by

DrRs:={uVs:ru(x)for a.e. xΩ, 0u(x)Rfor a.e. xΩ}.assignsuperscriptsubscript𝐷𝑟𝑅𝑠conditional-set𝑢superscript𝑉𝑠formulae-sequenceformulae-sequence𝑟𝑢𝑥formulae-sequencefor a.e. 𝑥superscriptΩ 0𝑢𝑥𝑅for a.e. 𝑥ΩD_{rR}^{s}:=\left\{u\in V^{s}:\ r\leq u\left(x\right)\ \ \text{for a.e. }x\in% \Omega^{\prime},\ 0\leq u\left(x\right)\leq R\ \ \text{for a.e. }x\in\Omega% \right\}.italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT := { italic_u ∈ italic_V start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT : italic_r ≤ italic_u ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 0 ≤ italic_u ( italic_x ) ≤ italic_R for a.e. italic_x ∈ roman_Ω } .

Hence, if uDrRs,𝑢superscriptsubscript𝐷𝑟𝑅𝑠u\in D_{rR}^{s},italic_u ∈ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT , then

u𝑢\displaystyle uitalic_u \displaystyle\in H01(Ω;2),divu=divus=0,superscriptsubscript𝐻01Ωsuperscript2div𝑢divsuperscript𝑢𝑠0\displaystyle H_{0}^{1}\left(\Omega;\mathbb{R}^{2}\right),\ \ \operatorname{% div}\,u=\operatorname{div}\,u^{s}=0,italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , roman_div italic_u = roman_div italic_u start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT = 0 ,
risubscript𝑟𝑖\displaystyle r_{i}italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT \displaystyle\leq ui(x)for a.e. xΩ, 0ui(x)Rifor a.e. xΩ(i=1,2).formulae-sequenceformulae-sequencesubscript𝑢𝑖𝑥for a.e. 𝑥superscriptΩ 0subscript𝑢𝑖𝑥subscript𝑅𝑖for a.e. 𝑥Ω𝑖12\displaystyle u_{i}\left(x\right)\ \text{for a.e. }x\in\Omega^{\prime},\ 0\leq u% _{i}\left(x\right)\leq R_{i}\ \text{for a.e. }x\in\Omega\ \ \left(i=1,2\right).italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , 0 ≤ italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_x ) ≤ italic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT for a.e. italic_x ∈ roman_Ω ( italic_i = 1 , 2 ) .

It is easy to see that the set DrRssuperscriptsubscript𝐷𝑟𝑅𝑠\ D_{rR}^{s}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT is closed in H01(Ω;2).superscriptsubscript𝐻01Ωsuperscript2H_{0}^{1}\left(\Omega;\mathbb{R}^{2}\right).italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .

Let us consider the matrix =(μij)1i,j2subscriptsubscript𝜇𝑖𝑗formulae-sequence1𝑖𝑗2\mathcal{M}=\left(\mu_{ij}\right)_{1\leq i,j\leq 2}caligraphic_M = ( italic_μ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 1 ≤ italic_i , italic_j ≤ 2 end_POSTSUBSCRIPT with entries

μij=lijλ1,subscript𝜇𝑖𝑗subscript𝑙𝑖𝑗subscript𝜆1\mu_{ij}=\frac{l_{ij}}{\lambda_{1}},italic_μ start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = divide start_ARG italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ,

where

lij=maxτ1[r1,R1]τ2[r2,R2]|ηiτj(τ1,τ2)|subscript𝑙𝑖𝑗subscriptsubscript𝜏1subscript𝑟1subscript𝑅1subscript𝜏2subscript𝑟2subscript𝑅2subscript𝜂𝑖subscript𝜏𝑗subscript𝜏1subscript𝜏2l_{ij}=\max_{{}_{\begin{subarray}{c}\tau_{1}\in\left[r_{1},R_{1}\right]\\ \tau_{2}\in\left[r_{2},R_{2}\right]\end{subarray}}}\left|\frac{\partial\eta_{i% }}{\partial\tau_{j}}\left(\tau_{1},\tau_{2}\right)\right|italic_l start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = roman_max start_POSTSUBSCRIPT start_FLOATSUBSCRIPT start_ARG start_ROW start_CELL italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ [ italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] end_CELL end_ROW start_ROW start_CELL italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ [ italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] end_CELL end_ROW end_ARG end_FLOATSUBSCRIPT end_POSTSUBSCRIPT | divide start_ARG ∂ italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_τ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) | (3.17)

and λ1subscript𝜆1\lambda_{1}italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the first eigenvalue of ΔΔ-\Delta- roman_Δ in Ω,Ω\Omega,roman_Ω , with the Dirichlet boundary condition.

For a function hL(Ω;+2)superscript𝐿Ωsuperscriptsubscript2\ h\in L^{\infty}\left(\Omega;\mathbb{R}_{+}^{2}\right)\ italic_h ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )denote

mhi=infΩhi,Mhi=hiL(Ω)(i=1,2).formulae-sequencesubscript𝑚subscript𝑖subscriptinfimumsuperscriptΩsubscript𝑖subscript𝑀subscript𝑖subscriptnormsubscript𝑖superscript𝐿Ω𝑖12m_{h_{i}}=\inf_{\Omega^{\prime}}h_{i},\ \ \ M_{h_{i}}=\left\|h_{i}\right\|_{L^% {\infty}\left(\Omega\right)}\ \ \left(i=1,2\right).italic_m start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_inf start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_M start_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ∥ italic_h start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ( italic_i = 1 , 2 ) .

Also, let us consider the following bounds of η𝜂\ \etaitalic_η given only locally:

mηi=minτ1[r1,R1]τ2[r2,R2]ηi(τ1,τ2),Mηi:=maxτ1[r1,R1]τ2[r2,R2]ηi(τ1,τ2)(i=1,2).formulae-sequencesubscript𝑚subscript𝜂𝑖subscriptsubscript𝜏1subscript𝑟1subscript𝑅1subscript𝜏2subscript𝑟2subscript𝑅2subscript𝜂𝑖subscript𝜏1subscript𝜏2assignsubscript𝑀subscript𝜂𝑖subscriptsubscript𝜏1subscript𝑟1subscript𝑅1subscript𝜏2subscript𝑟2subscript𝑅2subscript𝜂𝑖subscript𝜏1subscript𝜏2𝑖12m_{\eta_{i}}=\min_{{}_{\begin{subarray}{c}\tau_{1}\in\left[r_{1},R_{1}\right]% \\ \tau_{2}\in\left[r_{2},R_{2}\right]\end{subarray}}}\eta_{i}\left(\tau_{1},\tau% _{2}\right),\ \ \ \ M_{\eta_{i}}:=\max_{\begin{subarray}{c}\tau_{1}\in\left[r_% {1},R_{1}\right]\\ \tau_{2}\in\left[r_{2},R_{2}\right]\end{subarray}}\eta_{i}\left(\tau_{1},\tau_% {2}\right)\ \ \ \left(i=1,2\right).italic_m start_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_min start_POSTSUBSCRIPT start_FLOATSUBSCRIPT start_ARG start_ROW start_CELL italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ [ italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] end_CELL end_ROW start_ROW start_CELL italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ [ italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] end_CELL end_ROW end_ARG end_FLOATSUBSCRIPT end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , italic_M start_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT := roman_max start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ [ italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] end_CELL end_ROW start_ROW start_CELL italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ [ italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ( italic_i = 1 , 2 ) . (3.18)

The main result in this section is the following localization theorem, also able to produce multiple solutions, depending on the oscillatory properties of function η𝜂\etaitalic_η.

Theorem 11

Let fXs𝑓superscript𝑋𝑠f\in X^{s}italic_f ∈ italic_X start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT, h=ΔL1fL(Ω;+2)Δsuperscript𝐿1𝑓superscript𝐿Ωsuperscriptsubscript2h=-\Delta L^{-1}f\in L^{\infty}\left(\Omega;\mathbb{R}_{+}^{2}\right)italic_h = - roman_Δ italic_L start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ∈ italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) and let η𝜂\etaitalic_η satisfy assumptions (3.16). If in addition η(τ)0𝜂𝜏0\eta\left(\tau\right)\geq 0italic_η ( italic_τ ) ≥ 0 for τ0,𝜏0\tau\geq 0,italic_τ ≥ 0 ,

r(mh+mη)γ,(Mh+Mη)ΓRformulae-sequence𝑟subscript𝑚subscript𝑚𝜂𝛾subscript𝑀subscript𝑀𝜂Γ𝑅r\leq\left(m_{h}+m_{\eta}\right)\gamma,\ \ \ \left(M_{h}+M_{\eta}\right)\Gamma\leq Ritalic_r ≤ ( italic_m start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ) italic_γ , ( italic_M start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ) roman_Γ ≤ italic_R (3.19)

and the spectral radius of the matrix \mathcal{M}caligraphic_M is strictly less than one, then problem (3.11) has a unique solution (u,p)𝑢𝑝\left(u,p\right)( italic_u , italic_p ) in Dr,Rs×L2(Ω)superscriptsubscript𝐷𝑟𝑅𝑠superscript𝐿2ΩD_{r,R}^{s}\times L^{2}\left(\Omega\right)italic_D start_POSTSUBSCRIPT italic_r , italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT × italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) with p=0.𝑝0\nabla p=0.∇ italic_p = 0 .

Proof. We follow the same ideas as in the proof of Theorem 3. Let vDrRs.𝑣superscriptsubscript𝐷𝑟𝑅𝑠\ v\in D_{rR}^{s}.italic_v ∈ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT . Since rv(x)R𝑟𝑣𝑥𝑅\ r\leq v\left(x\right)\leq Ritalic_r ≤ italic_v ( italic_x ) ≤ italic_R on ΩsuperscriptΩ\Omega^{\prime}roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and 0v(x) 0𝑣𝑥\ 0\leq v\left(x\right)0 ≤ italic_v ( italic_x ) on Ω,Ω\Omega,roman_Ω , we have 01Ω(x)η(v(x))Mη 0subscript1superscriptΩ𝑥𝜂𝑣𝑥subscript𝑀𝜂\ 0\leq 1_{\Omega^{\prime}}\left(x\right)\eta\left(v\left(x\right)\right)\leq M% _{\eta}\ \ 0 ≤ 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) italic_η ( italic_v ( italic_x ) ) ≤ italic_M start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPTa.e. in Ω.Ω\Omega.roman_Ω . Then using the positivity and monotonicity properties of (Δ)1,superscriptΔ1\left(-\Delta\right)^{-1},( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , we obtain

00\displaystyle 0 \displaystyle\leq T(v)(x)=(Δ)1(h+1Ωη(v))(x)𝑇𝑣𝑥superscriptΔ1subscript1superscriptΩ𝜂𝑣𝑥\displaystyle T\left(v\right)\left(x\right)=\left(-\Delta\right)^{-1}\left(h+1% _{\Omega^{\prime}}\eta\left(v\right)\right)\left(x\right)italic_T ( italic_v ) ( italic_x ) = ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_h + 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_η ( italic_v ) ) ( italic_x )
\displaystyle\leq (Mh+Mη)((Δ)11)(x)(Mh+Mη)Γfor a.e. xΩ.formulae-sequencesubscript𝑀subscript𝑀𝜂superscriptΔ11𝑥subscript𝑀subscript𝑀𝜂Γfor a.e. 𝑥Ω\displaystyle\left(M_{h}+M_{\eta}\right)\left(\left(-\Delta\right)^{-1}1\right% )\left(x\right)\leq\left(M_{h}+M_{\eta}\right)\Gamma\ \ \ \text{for a.e. }x\in\Omega.( italic_M start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ) ( ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT 1 ) ( italic_x ) ≤ ( italic_M start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ) roman_Γ for a.e. italic_x ∈ roman_Ω .

For the lower estimation in Ω,superscriptΩ\Omega^{\prime},roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , since rv(x)R𝑟𝑣𝑥𝑅\ r\leq v\left(x\right)\leq Ritalic_r ≤ italic_v ( italic_x ) ≤ italic_R on Ω,superscriptΩ\Omega^{\prime},roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , we have mηη(v(x))subscript𝑚𝜂𝜂𝑣𝑥\ m_{\eta}\leq\eta\left(v\left(x\right)\right)\ \ italic_m start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ≤ italic_η ( italic_v ( italic_x ) )a.e. in ΩsuperscriptΩ\Omega^{\prime}roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and thus 1Ω(x)mηη(v(x))subscript1superscriptΩ𝑥subscript𝑚𝜂𝜂𝑣𝑥\ 1_{\Omega^{\prime}}\left(x\right)m_{\eta}\leq\eta\left(v\left(x\right)\right% )\ \ 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) italic_m start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ≤ italic_η ( italic_v ( italic_x ) )a.e. in Ω.Ω\Omega.roman_Ω . Then

T(v)(x)(mh+mη)((Δ)11Ω)(x)(mh+mη)γfor a.e. xΩ.formulae-sequence𝑇𝑣𝑥subscript𝑚subscript𝑚𝜂superscriptΔ1subscript1superscriptΩ𝑥subscript𝑚subscript𝑚𝜂𝛾for a.e. 𝑥superscriptΩT\left(v\right)\left(x\right)\geq\left(m_{h}+m_{\eta}\right)\left(\left(-% \Delta\right)^{-1}1_{\Omega^{\prime}}\right)\left(x\right)\geq\left(m_{h}+m_{% \eta}\right)\gamma\ \ \text{for a.e. }x\in\Omega^{\prime}.italic_T ( italic_v ) ( italic_x ) ≥ ( italic_m start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ) ( ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) ( italic_x ) ≥ ( italic_m start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + italic_m start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ) italic_γ for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT . (3.20)

Hence T(DrRs)DrRs𝑇superscriptsubscript𝐷𝑟𝑅𝑠superscriptsubscript𝐷𝑟𝑅𝑠T\left(D_{rR}^{s}\right)\subset D_{rR}^{s}italic_T ( italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ) ⊂ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT due to (3.19).

Now we prove that the operator T𝑇Titalic_T satisfies on DrRssuperscriptsubscript𝐷𝑟𝑅𝑠D_{rR}^{s}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT the contraction condition in the sense of Perov. To this aim, let v1,v2DrRs,superscript𝑣1superscript𝑣2superscriptsubscript𝐷𝑟𝑅𝑠v^{1},v^{2}\in D_{rR}^{s},italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∈ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT , and denote wi=T(vi),superscript𝑤𝑖𝑇superscript𝑣𝑖w^{i}=T\left(v^{i}\right),italic_w start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_T ( italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) , i=1,2.𝑖12i=1,2.italic_i = 1 , 2 . Then

{Δwi=h+1Ωη(vi)in Ωwi=0on ΩcasesΔsuperscript𝑤𝑖subscript1superscriptΩ𝜂superscript𝑣𝑖in Ωsuperscript𝑤𝑖0on Ω\left\{\begin{array}[]{ll}-\Delta w^{i}=h+1_{\Omega^{\prime}}\eta\left(v^{i}% \right)&\text{in }\Omega\\ w^{i}=0&\text{on }\partial\Omega\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_Δ italic_w start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = italic_h + 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_η ( italic_v start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_w start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω end_CELL end_ROW end_ARRAY

in the weak sense, for i=1,2.𝑖12i=1,2.italic_i = 1 , 2 . These give

{Δ(w1w2)=(η(v1)η(v2))1Ωin Ωw1w2=0on ΩcasesΔsuperscript𝑤1superscript𝑤2𝜂superscript𝑣1𝜂superscript𝑣2subscript1superscriptΩin Ωsuperscript𝑤1superscript𝑤20on Ω\left\{\begin{array}[]{ll}-\Delta\left(w^{1}-w^{2}\right)=\left(\eta\left(v^{1% }\right)\ -\eta\left(v^{2}\right)\right)1_{\Omega^{\prime}}&\text{in }\Omega\\ w^{1}-w^{2}=0&\text{on }\partial\Omega\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_Δ ( italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = ( italic_η ( italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_η ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω end_CELL end_ROW end_ARRAY

in the weak sense. Thus, for each i=1,2,𝑖12i=1,2,italic_i = 1 , 2 , multiplying by wi1wi2superscriptsubscript𝑤𝑖1superscriptsubscript𝑤𝑖2w_{i}^{1}-w_{i}^{2}italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and integrating over ΩΩ\Omegaroman_Ω yield

wi1wi2H01(Ω)2superscriptsubscriptnormsuperscriptsubscript𝑤𝑖1superscriptsubscript𝑤𝑖2superscriptsubscript𝐻01Ω2\displaystyle\left\|w_{i}^{1}-w_{i}^{2}\right\|_{H_{0}^{1}\left(\Omega\right)}% ^{2}∥ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (3.21)
=\displaystyle== Ω(ηi(v1)ηi(v2))(wi1wi2)subscriptsuperscriptΩsubscript𝜂𝑖superscript𝑣1subscript𝜂𝑖superscript𝑣2superscriptsubscript𝑤𝑖1superscriptsubscript𝑤𝑖2\displaystyle\int_{\Omega^{\prime}}\left(\eta_{i}\left(v^{1}\right)-\eta_{i}% \left(v^{2}\right)\right)\left(w_{i}^{1}-w_{i}^{2}\right)∫ start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ( italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) (3.22)
\displaystyle\leq ηi(v1)ηi(v2)L2(Ω)wi1wi2L2(Ω)subscriptnormsubscript𝜂𝑖superscript𝑣1subscript𝜂𝑖superscript𝑣2superscript𝐿2superscriptΩsubscriptnormsuperscriptsubscript𝑤𝑖1superscriptsubscript𝑤𝑖2superscript𝐿2superscriptΩ\displaystyle\left\|\eta_{i}\left(v^{1}\right)-\eta_{i}\left(v^{2}\right)% \right\|_{L^{2}\left(\Omega^{\prime}\right)}\left\|w_{i}^{1}-w_{i}^{2}\right\|% _{L^{2}\left(\Omega^{\prime}\right)}∥ italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT ∥ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT (3.23)
\displaystyle\leq (ηi(v11,v21)ηi(v12,v21)L2(Ω)+ηi(v12,v21)ηi(v12,v22)L2(Ω))wi1wi2L2(Ω)subscriptnormsubscript𝜂𝑖superscriptsubscript𝑣11superscriptsubscript𝑣21subscript𝜂𝑖superscriptsubscript𝑣12superscriptsubscript𝑣21superscript𝐿2superscriptΩsubscriptnormsubscript𝜂𝑖superscriptsubscript𝑣12superscriptsubscript𝑣21subscript𝜂𝑖superscriptsubscript𝑣12superscriptsubscript𝑣22superscript𝐿2superscriptΩsubscriptnormsuperscriptsubscript𝑤𝑖1superscriptsubscript𝑤𝑖2superscript𝐿2Ω\displaystyle\left(\left\|\eta_{i}\left(v_{1}^{1},v_{2}^{1}\right)-\eta_{i}% \left(v_{1}^{2},v_{2}^{1}\right)\right\|_{L^{2}\left(\Omega^{\prime}\right)}+% \left\|\eta_{i}\left(v_{1}^{2},v_{2}^{1}\right)-\eta_{i}\left(v_{1}^{2},v_{2}^% {2}\right)\right\|_{L^{2}\left(\Omega^{\prime}\right)}\right)\left\|w_{i}^{1}-% w_{i}^{2}\right\|_{L^{2}\left(\Omega\right)}( ∥ italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT + ∥ italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT ) ∥ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT (3.24)

One has

ηi(v11,v21)ηi(v12,v21)L2(Ω)subscriptnormsubscript𝜂𝑖superscriptsubscript𝑣11superscriptsubscript𝑣21subscript𝜂𝑖superscriptsubscript𝑣12superscriptsubscript𝑣21superscript𝐿2superscriptΩ\displaystyle\left\|\eta_{i}\left(v_{1}^{1},v_{2}^{1}\right)-\eta_{i}\left(v_{% 1}^{2},v_{2}^{1}\right)\right\|_{L^{2}\left(\Omega^{\prime}\right)}∥ italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT \displaystyle\leq li1v11v12L2(Ω),subscript𝑙𝑖1subscriptnormsuperscriptsubscript𝑣11superscriptsubscript𝑣12superscript𝐿2Ω\displaystyle l_{i1}\left\|v_{1}^{1}-v_{1}^{2}\right\|_{L^{2}\left(\Omega% \right)},italic_l start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT ∥ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ,
ηi(v12,v21)ηi(v12,v22)L2(Ω)subscriptnormsubscript𝜂𝑖superscriptsubscript𝑣12superscriptsubscript𝑣21subscript𝜂𝑖superscriptsubscript𝑣12superscriptsubscript𝑣22superscript𝐿2superscriptΩ\displaystyle\left\|\eta_{i}\left(v_{1}^{2},v_{2}^{1}\right)-\eta_{i}\left(v_{% 1}^{2},v_{2}^{2}\right)\right\|_{L^{2}\left(\Omega^{\prime}\right)}∥ italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT \displaystyle\leq li2v21v22L2(Ω).subscript𝑙𝑖2subscriptnormsuperscriptsubscript𝑣21superscriptsubscript𝑣22superscript𝐿2Ω\displaystyle l_{i2}\left\|v_{2}^{1}-v_{2}^{2}\right\|_{L^{2}\left(\Omega% \right)}.italic_l start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT ∥ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT .

Using twice Poincaré’s inequality we deduce

wi1wi2H01(Ω)1λ1(li1v11v12H01(Ω)+li2v21v22H01(Ω)).subscriptnormsuperscriptsubscript𝑤𝑖1superscriptsubscript𝑤𝑖2superscriptsubscript𝐻01Ω1subscript𝜆1subscript𝑙𝑖1subscriptnormsuperscriptsubscript𝑣11superscriptsubscript𝑣12superscriptsubscript𝐻01Ωsubscript𝑙𝑖2subscriptnormsuperscriptsubscript𝑣21superscriptsubscript𝑣22superscriptsubscript𝐻01Ω\left\|w_{i}^{1}-w_{i}^{2}\right\|_{H_{0}^{1}\left(\Omega\right)}\leq\frac{1}{% \lambda_{1}}\left(l_{i1}\left\|v_{1}^{1}-v_{1}^{2}\right\|_{H_{0}^{1}\left(% \Omega\right)}+l_{i2}\left\|v_{2}^{1}-v_{2}^{2}\right\|_{H_{0}^{1}\left(\Omega% \right)}\right).∥ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ( italic_l start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT ∥ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT + italic_l start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT ∥ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT ) .

Therefore

(T1(v1)T1(v2)H01(Ω)T2(v1)T2(v2)H01(Ω))(v11v12H01(Ω)v21v22H01(Ω)).subscriptnormsubscript𝑇1superscript𝑣1subscript𝑇1superscript𝑣2superscriptsubscript𝐻01Ωsubscriptnormsubscript𝑇2superscript𝑣1subscript𝑇2superscript𝑣2superscriptsubscript𝐻01Ωsubscriptnormsuperscriptsubscript𝑣11superscriptsubscript𝑣12superscriptsubscript𝐻01Ωsubscriptnormsuperscriptsubscript𝑣21superscriptsubscript𝑣22superscriptsubscript𝐻01Ω\left(\begin{array}[]{c}\left\|T_{1}\left(v^{1}\right)-T_{1}\left(v^{2}\right)% \right\|_{H_{0}^{1}\left(\Omega\right)}\\ \left\|T_{2}\left(v^{1}\right)-T_{2}\left(v^{2}\right)\right\|_{H_{0}^{1}\left% (\Omega\right)}\end{array}\right)\leq\mathcal{M}\left(\begin{array}[]{c}\left% \|v_{1}^{1}-v_{1}^{2}\right\|_{H_{0}^{1}\left(\Omega\right)}\\ \left\|v_{2}^{1}-v_{2}^{2}\right\|_{H_{0}^{1}\left(\Omega\right)}\end{array}% \right).( start_ARRAY start_ROW start_CELL ∥ italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ∥ italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) - italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) ≤ caligraphic_M ( start_ARRAY start_ROW start_CELL ∥ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ∥ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT - italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( roman_Ω ) end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) .

Since the spectral radius of \mathcal{M}caligraphic_M is less than one, we may apply Perov’s fixed point theorem to deduce the existence and uniqueness of uDrRs𝑢superscriptsubscript𝐷𝑟𝑅𝑠u\in D_{rR}^{s}italic_u ∈ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT with u=T(u).𝑢𝑇𝑢u=T\left(u\right).italic_u = italic_T ( italic_u ) .   

Remark 12

If the functions η1,η2subscript𝜂1subscript𝜂2\eta_{1},\eta_{2}italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are nondecreasing in both variables in [r1,R1]×[r2,R2],subscript𝑟1subscript𝑅1subscript𝑟2subscript𝑅2\left[r_{1},R_{1}\right]\times\left[r_{2},R_{2}\right],[ italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] × [ italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] , then  mηi=ηi(r),subscript𝑚subscript𝜂𝑖subscript𝜂𝑖𝑟m_{\eta_{i}}=\eta_{i}\left(r\right),italic_m start_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_r ) , Mηi=ηi(R)subscript𝑀subscript𝜂𝑖subscript𝜂𝑖𝑅M_{\eta_{i}}=\eta_{i}\left(R\right)italic_M start_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_R ) for i=1,2𝑖12i=1,2italic_i = 1 , 2 and condition (3.19) becomes

r(mh+η(r))γ,(Mh+η(R))ΓR.formulae-sequence𝑟subscript𝑚𝜂𝑟𝛾subscript𝑀𝜂𝑅Γ𝑅r\leq\left(m_{h}+\eta\left(r\right)\right)\gamma,\ \ \ \ \left(M_{h}+\eta(R)% \right)\Gamma\leq R.\vskip 6.0pt plus 2.0pt minus 2.0ptitalic_r ≤ ( italic_m start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + italic_η ( italic_r ) ) italic_γ , ( italic_M start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT + italic_η ( italic_R ) ) roman_Γ ≤ italic_R .

Let us notice that such vectors  r,R𝑟𝑅r,R\ italic_r , italic_R always exist if the asymptotic conditions below are satisfied:

limτi0ηi(τ1,τ2)τi=+,limτi+ηi(τ1,τ2)τi=0(i=1,2).formulae-sequencesubscriptsubscript𝜏𝑖0subscript𝜂𝑖subscript𝜏1subscript𝜏2subscript𝜏𝑖subscriptsubscript𝜏𝑖subscript𝜂𝑖subscript𝜏1subscript𝜏2subscript𝜏𝑖0𝑖12\ \lim_{\tau_{i}\rightarrow 0}\frac{\eta_{i}\left(\tau_{1},\tau_{2}\right)}{% \tau_{i}}=+\infty,\ \ \ \lim_{\tau_{i}\rightarrow+\infty}\frac{\eta_{i}\left(% \tau_{1},\tau_{2}\right)}{\tau_{i}}=0\ \ \left(i=1,2\right).roman_lim start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → 0 end_POSTSUBSCRIPT divide start_ARG italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = + ∞ , roman_lim start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → + ∞ end_POSTSUBSCRIPT divide start_ARG italic_η start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_τ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = 0 ( italic_i = 1 , 2 ) .

One also has the following variational characterizations of the localized solution.

Theorem 13

Under the assumptions of Theorem 11,

(a) u=(u1,u2)𝑢subscript𝑢1subscript𝑢2u=\left(u_{1},u_{2}\right)italic_u = ( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) is a Nash equilibrium in DrR ssuperscriptsubscript𝐷𝑟𝑅 𝑠D_{rR\text{ }}^{s}\ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPTfor the pair of partial energy functionals

J1(v1,v2)subscript𝐽1subscript𝑣1subscript𝑣2\displaystyle J_{1}\left(v_{1},v_{2}\right)italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) =\displaystyle== Ω(12|v1|2h1v1)𝑑xΩ0v1(x)η1(τ,v2(x))𝑑τ𝑑x,subscriptΩ12superscriptsubscript𝑣12subscript1subscript𝑣1differential-d𝑥subscriptsuperscriptΩsuperscriptsubscript0subscript𝑣1𝑥subscript𝜂1𝜏subscript𝑣2𝑥differential-d𝜏differential-d𝑥\displaystyle\int_{\Omega}\left(\frac{1}{2}\left|\nabla v_{1}\right|^{2}-h_{1}% v_{1}\right)dx-\int_{\Omega^{\prime}}\int_{0}^{v_{1}\left(x\right)}\eta_{1}% \left(\tau,v_{2}\left(x\right)\right)d\tau dx,∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG | ∇ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_h start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_d italic_x - ∫ start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ) end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_τ , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) ) italic_d italic_τ italic_d italic_x ,
J2(v1,v2)subscript𝐽2subscript𝑣1subscript𝑣2\displaystyle J_{2}\left(v_{1},v_{2}\right)italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) =\displaystyle== Ω(12|v2|2h2v2)𝑑xΩ0v2(x)η2(v1(x),τ)𝑑τ𝑑x.subscriptΩ12superscriptsubscript𝑣22subscript2subscript𝑣2differential-d𝑥subscriptsuperscriptΩsuperscriptsubscript0subscript𝑣2𝑥subscript𝜂2subscript𝑣1𝑥𝜏differential-d𝜏differential-d𝑥\displaystyle\int_{\Omega}\left(\frac{1}{2}\left|\nabla v_{2}\right|^{2}-h_{2}% v_{2}\right)dx-\int_{\Omega^{\prime}}\int_{0}^{v_{2}\left(x\right)}\eta_{2}% \left(v_{1}\left(x\right),\tau\right)d\tau dx.∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG | ∇ italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_h start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_d italic_x - ∫ start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ) , italic_τ ) italic_d italic_τ italic_d italic_x .

(b) if in addition η𝜂\etaitalic_η is of potential type, i.e., η=H,𝜂𝐻\ \eta=\nabla H,italic_η = ∇ italic_H , then u𝑢uitalic_u minimizes in

DrRssuperscriptsubscript𝐷𝑟𝑅𝑠D_{rR}^{s}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT the functional

JD(v)=Ω(12|v|2hv)𝑑xΩH(v)𝑑x.subscript𝐽𝐷𝑣subscriptΩ12superscript𝑣2𝑣differential-d𝑥subscriptsuperscriptΩ𝐻𝑣differential-d𝑥J_{D}\left(v\right)=\int_{\Omega}\left(\frac{1}{2}\left|\nabla v\right|^{2}-h% \cdot v\right)dx-\int_{\Omega^{\prime}}H\left(v\right)dx.italic_J start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT ( italic_v ) = ∫ start_POSTSUBSCRIPT roman_Ω end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG | ∇ italic_v | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_h ⋅ italic_v ) italic_d italic_x - ∫ start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_H ( italic_v ) italic_d italic_x .

Proof. The proof is similar to that of Theorem 4.1 in [4] and we omit it.   

4 Stokes systems with variable periodic viscosity

This section is devoted to the Stokes system with variable periodic viscosity (see [1]). We follow the same goals as in the previous section, namely the localization and multiplicity of solutions. The physical motivation consists in the incompressible flow in a domain filled with a mixture of fluids having a highly heterogeneous viscosity denoted με(x),superscript𝜇𝜀𝑥\mu^{\varepsilon}\left(x\right),italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) , which is a periodically varying function of the space variable with small period ε>0𝜀0\varepsilon>0italic_ε > 0. Mathematically, we have a family of equations depending on the small parameter ε,𝜀\varepsilon,italic_ε , and the problem consists in an asymptotic analysis, as ε0,𝜀0\varepsilon\rightarrow 0,italic_ε → 0 , aimed to give an averaged description of the process. As before, let fL2(Ω;n)𝑓superscript𝐿2Ωsuperscript𝑛f\in L^{2}(\Omega;{\mathbb{R}}^{n})italic_f ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) be a function representing the given external force. Let Y=(0,1)n𝑌superscript01𝑛Y=(0,1)^{n}italic_Y = ( 0 , 1 ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be the unitary n𝑛nitalic_n-cube and μ:Y:𝜇𝑌\mu:Y\longrightarrow{\mathbb{R}}italic_μ : italic_Y ⟶ blackboard_R a Y𝑌Yitalic_Y-periodic function belonging to L(Y)superscript𝐿𝑌L^{\infty}(Y)italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_Y ). We define by με(x)=μ(xε)superscript𝜇𝜀𝑥𝜇𝑥𝜀\mu^{\varepsilon}(x)=\mu\left(\frac{x}{\varepsilon}\right)italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) = italic_μ ( divide start_ARG italic_x end_ARG start_ARG italic_ε end_ARG ) the ε𝜀\varepsilonitalic_ε-periodic scaled function, which represents the viscosity of a complex mixture of fluids having different viscosities. We assume that the viscosity με(x)superscript𝜇𝜀𝑥\mu^{\varepsilon}\left(x\right)italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) is such that μεC1(Ω¯)superscript𝜇𝜀superscript𝐶1¯Ω\mu^{\varepsilon}\in C^{1}\left(\overline{\Omega}\right)italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( over¯ start_ARG roman_Ω end_ARG ) and

0<μ¯με(x)μ¯ for all xΩ. formulae-sequence0¯𝜇superscript𝜇𝜀𝑥¯𝜇 for all 𝑥Ω 0<\underline{\mu}\leq\mu^{\varepsilon}\left(x\right)\leq\overline{\mu}\text{\ % \ \ for all }x\in\Omega.\text{ }0 < under¯ start_ARG italic_μ end_ARG ≤ italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ≤ over¯ start_ARG italic_μ end_ARG for all italic_x ∈ roman_Ω . (4.1)

Note that according to remarks (R2), (R3) from Section 2 and in view of (4.1), the constants ΓfsubscriptΓ𝑓\Gamma_{f}roman_Γ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT and γfsubscript𝛾𝑓\gamma_{f}italic_γ start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT do not depend on ε.𝜀\varepsilon.italic_ε . Consequently, all the estimations in Section 3 remain valid and we immediately obtain the following versions of the results in Section 3, for the ε𝜀\varepsilonitalic_ε-parametrized problems and the homogenized ones. The convergence results and the localization of the solution for the homogenized problems are obtained as in [4, Theorem 3.2].

Thus we discuss below a series of parametrized problems.

1a. We start with the analogue of system (2.1), with the given external force f𝑓fitalic_f, namely

{div(με(x)uε)+pε=f(x)in Ωdivuε=0in Ωuε=0on Ω,casesdivsuperscript𝜇𝜀𝑥superscript𝑢𝜀superscript𝑝𝜀𝑓𝑥in Ωdivsuperscript𝑢𝜀0in Ωsuperscript𝑢𝜀0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu^{\varepsilon}\left(x% \right)\nabla u^{\varepsilon}\right)+\nabla p^{\varepsilon}=f\left(x\right)&% \text{in }\Omega\\ \operatorname{div}\,u^{\varepsilon}=0&\text{in }\Omega\\ u^{\varepsilon}=0&\text{on }\partial\Omega,\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ∇ italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) + ∇ italic_p start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = italic_f ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω , end_CELL end_ROW end_ARRAY (4.2)

where uεsuperscript𝑢𝜀u^{\varepsilon}italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT and pεsuperscript𝑝𝜀p^{\varepsilon}italic_p start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT represent the velocity and the pressure of the mixture. For a fixed ε𝜀\varepsilonitalic_ε, system (4.2) is equivalent, as in the nonparametric case, to the diffusion problem

{div(με(x)wε)=f(x)in Ωwε=0on Ω.casesdivsuperscript𝜇𝜀𝑥superscript𝑤𝜀𝑓𝑥in Ωsuperscript𝑤𝜀0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu^{\varepsilon}\left(x% \right)\nabla w^{\varepsilon}\right)=f\left(x\right)&\text{in }\Omega\\ w^{\varepsilon}=0&\text{on }\partial\Omega.\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ∇ italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) = italic_f ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω . end_CELL end_ROW end_ARRAY (4.3)

The following analogue of Proposition 1 holds.

Theorem 14

If

rγ1Ωf(x)for a.e. xΩand 0f(x)RΓ1for a.e. xΩ,formulae-sequenceformulae-sequence𝑟subscript𝛾subscript1superscriptΩ𝑓𝑥formulae-sequencefor a.e. 𝑥superscriptΩand 0𝑓𝑥𝑅subscriptΓ1for a.e. 𝑥Ω\frac{r}{\gamma_{1_{\Omega^{\prime}}}}\leq f\left(x\right)\ \ \ \text{for a.e.% }x\in\Omega^{\prime}\ \ \ \text{and\ \ \ }0\leq f\left(x\right)\leq\frac{R}{% \Gamma_{1}}\ \ \text{for a.e. }x\in\Omega,divide start_ARG italic_r end_ARG start_ARG italic_γ start_POSTSUBSCRIPT 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG ≤ italic_f ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and 0 ≤ italic_f ( italic_x ) ≤ divide start_ARG italic_R end_ARG start_ARG roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG for a.e. italic_x ∈ roman_Ω ,

then the solution wε=(Lε)1fsuperscript𝑤𝜀superscriptsuperscript𝐿𝜀1𝑓w^{\varepsilon}=\left(L^{\varepsilon}\right)^{-1}fitalic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = ( italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f   of (4.3) belongs to DrRsubscript𝐷𝑟𝑅D_{rR}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT. There is a function w0H01(Ω;n)superscript𝑤0subscriptsuperscript𝐻10Ωsuperscript𝑛w^{0}\in H^{1}_{0}(\Omega;{\mathbb{R}}^{n})italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) such that

wεw0weakly in H01(Ω;n),superscript𝑤𝜀superscript𝑤0weakly in subscriptsuperscript𝐻10Ωsuperscript𝑛w^{\varepsilon}\rightharpoonup w^{0}\quad\quad\text{weakly in }H^{1}_{0}(% \Omega;{\mathbb{R}}^{n}),italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ⇀ italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT weakly in italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ,

w0superscript𝑤0w^{0}italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT belongs to DrRsubscript𝐷𝑟𝑅D_{rR}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT and it is the unique solution of the following homogenized problem

{div(Ahomw0)=fin Ωw0=0on Ω,casesdivsuperscript𝐴homsuperscript𝑤0𝑓in Ωsuperscript𝑤00on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(A^{\text{hom}}\nabla w^{0}% \right)=f&\text{in }\Omega\\ w^{0}=0&\text{on }\partial\Omega,\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT ∇ italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) = italic_f end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_w start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω , end_CELL end_ROW end_ARRAY (4.4)

where the entries of the positive definite homogenized matrix Ahomsuperscript𝐴homA^{\text{hom}}italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT are given for i,j{1,..,n}i,j\in\left\{1,..,n\right\}italic_i , italic_j ∈ { 1 , . . , italic_n } by

Aijhom=Yμ(y)(χjyj)(χiyi)dy,subscriptsuperscript𝐴hom𝑖𝑗subscript𝑌𝜇𝑦superscript𝜒𝑗subscript𝑦𝑗superscript𝜒𝑖subscript𝑦𝑖𝑑𝑦A^{\text{hom}}_{ij}=\int\limits_{Y}\mu(y)\nabla(\chi^{j}-y_{j})\cdot\nabla(% \chi^{i}-y_{i})dy,italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = ∫ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT italic_μ ( italic_y ) ∇ ( italic_χ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ⋅ ∇ ( italic_χ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_d italic_y , (4.5)

and χjHper1(Y)={vH1(Y);v is Yperiodic}superscript𝜒𝑗subscriptsuperscript𝐻1per𝑌𝑣superscript𝐻1𝑌𝑣 is 𝑌periodic\chi^{j}\in H^{1}_{\text{per}}(Y)=\{v\in H^{1}(Y);\,v\text{ is }\,Y-\text{% periodic}\}italic_χ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT per end_POSTSUBSCRIPT ( italic_Y ) = { italic_v ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_Y ) ; italic_v is italic_Y - periodic } are the unique solutions of the local problems

{divy(μ(y)y(χjyj)=0in YYχj𝑑y=0.\left\{\begin{array}[]{ll}-\operatorname{div}_{y}\left(\mu\left(y\right)\nabla% _{y}(\chi^{j}-y_{j}\right)=0&\text{in }Y\\ \int\limits_{Y}\chi^{j}dy=0.&\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_μ ( italic_y ) ∇ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_χ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = 0 end_CELL start_CELL in italic_Y end_CELL end_ROW start_ROW start_CELL ∫ start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT italic_χ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_d italic_y = 0 . end_CELL start_CELL end_CELL end_ROW end_ARRAY (4.6)

1b. We consider now the analogue of system (2.11), with an ε𝜀\varepsilonitalic_ε-dependent reaction force, namely

{div(με(x)uε)+pε=h(x,uε(x)+qε(x))in Ωdivuε=0in Ωuε=0on Ω.casesdivsuperscript𝜇𝜀𝑥superscript𝑢𝜀superscript𝑝𝜀𝑥superscript𝑢𝜀𝑥superscript𝑞𝜀𝑥in Ωdivsuperscript𝑢𝜀0in Ωsuperscript𝑢𝜀0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu^{\varepsilon}\left(x% \right)\nabla u^{\varepsilon}\right)+\nabla p^{\varepsilon}=h\left(x,u^{% \varepsilon}\left(x\right)+q^{\varepsilon}\left(x\right)\right)&\text{in }% \Omega\\ \operatorname{div}\,u^{\varepsilon}=0&\text{in }\Omega\\ u^{\varepsilon}=0&\text{on }\partial\Omega.\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ∇ italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) + ∇ italic_p start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = italic_h ( italic_x , italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) + italic_q start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω . end_CELL end_ROW end_ARRAY (4.7)

For a fixed ε𝜀\varepsilonitalic_ε, system (4.7) is equivalent with the diffusion problem

{div(με(x)wε)=h(x,wε(x))in Ωwε=0on Ω.casesdivsuperscript𝜇𝜀𝑥superscript𝑤𝜀𝑥superscript𝑤𝜀𝑥in Ωsuperscript𝑤𝜀0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu^{\varepsilon}\left(x% \right)\nabla w^{\varepsilon}\right)=h\left(x,w^{\varepsilon}\left(x\right)% \right)&\text{in }\Omega\\ w^{\varepsilon}=0&\text{on }\partial\Omega.\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ∇ italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) = italic_h ( italic_x , italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω . end_CELL end_ROW end_ARRAY (4.8)

Let the operator Lεsuperscript𝐿𝜀L^{\varepsilon}italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT be defined by Lεv=div(με(x)v))L^{\varepsilon}v=-\operatorname{div}(\mu^{\varepsilon}(x)\nabla v))italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT italic_v = - roman_div ( italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ∇ italic_v ) ). The analogue of Theorem 3 is the following.

Theorem 15

Assume that hhitalic_h satisfies (h1) and (h2) with

r(Lε)1h¯ on Ωand (Lε)1h¯Ron Ωformulae-sequence𝑟superscriptsuperscript𝐿𝜀1¯ on superscriptΩand superscriptsuperscript𝐿𝜀1¯𝑅on Ωr\leq\left(L^{\varepsilon}\right)^{-1}\underline{h}\text{ \ \ on\ }\Omega^{% \prime}\ \ \ \text{and\ \ \ }\left(L^{\varepsilon}\right)^{-1}\overline{h}\leq R% \ \ \ \text{on\ }\Omegaitalic_r ≤ ( italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT under¯ start_ARG italic_h end_ARG on roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and ( italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over¯ start_ARG italic_h end_ARG ≤ italic_R on roman_Ω

for every ε(0,ε0]𝜀0subscript𝜀0\varepsilon\in(0,\varepsilon_{0}]italic_ε ∈ ( 0 , italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] and some ε0>0.subscript𝜀00\varepsilon_{0}>0.italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 . Then problem (4.8) has a unique solution wεDrR.superscript𝑤𝜀subscript𝐷𝑟𝑅w^{\varepsilon}\in D_{rR}.italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ∈ italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT . Moreover, wεsuperscript𝑤𝜀w^{\varepsilon}italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT is in DrRsubscript𝐷𝑟𝑅D_{rR}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT a Nash equilibrium with respect to the partial energy functionals of the system. There is a function w1H01(Ω;n)superscript𝑤1subscriptsuperscript𝐻10Ωsuperscript𝑛w^{1}\in H^{1}_{0}(\Omega;{\mathbb{R}}^{n})italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) such that

wεw1weakly in H01(Ω;n),superscript𝑤𝜀superscript𝑤1weakly in subscriptsuperscript𝐻10Ωsuperscript𝑛w^{\varepsilon}\rightharpoonup w^{1}\quad\quad\text{weakly in }H^{1}_{0}(% \Omega;{\mathbb{R}}^{n}),italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ⇀ italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT weakly in italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) ,

w1superscript𝑤1w^{1}italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT belongs to DrRsubscript𝐷𝑟𝑅D_{rR}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT and it is the unique solution of the homogenized problem

{div(Ahomw1)=h(x,w1)in Ωw1=0on Ω,casesdivsuperscript𝐴homsuperscript𝑤1𝑥superscript𝑤1in Ωsuperscript𝑤10on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(A^{\text{hom}}\,\nabla w^{1% }\right)=h(x,w^{1})&\text{in }\Omega\\ w^{1}=0&\text{on }\partial\Omega,\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT ∇ italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) = italic_h ( italic_x , italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω , end_CELL end_ROW end_ARRAY (4.9)

where the matrix Ahomsuperscript𝐴homA^{\text{hom}}italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT is defined in (4.5).

So for both problems (4.2) and (4.7) we localized the recovered velocity wεsuperscript𝑤𝜀\ w^{\varepsilon}italic_w start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT, as far as the homogenized recovered velocities w𝑤witalic_w and w1superscript𝑤1w^{1}italic_w start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT, respectively.

2a. The analogue of system (2.1), with an ε𝜀\varepsilonitalic_ε-dependent external force is

{div(με(x)uε)+pε=fε(x)in Ωdivuε=0in Ωuε=0on Ω,casesdivsuperscript𝜇𝜀𝑥superscript𝑢𝜀superscript𝑝𝜀superscript𝑓𝜀𝑥in Ωdivsuperscript𝑢𝜀0in Ωsuperscript𝑢𝜀0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu^{\varepsilon}\left(x% \right)\nabla u^{\varepsilon}\right)+\nabla p^{\varepsilon}=f^{\varepsilon}% \left(x\right)&\text{in }\Omega\\ \operatorname{div}\,u^{\varepsilon}=0&\text{in }\Omega\\ u^{\varepsilon}=0&\text{on }\partial\Omega,\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ∇ italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) + ∇ italic_p start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = italic_f start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω , end_CELL end_ROW end_ARRAY (4.10)

where divΔ(Lε)1fε=0;divΔsuperscriptsuperscript𝐿𝜀1superscript𝑓𝜀0\ \operatorname{div}\Delta\left(L^{\varepsilon}\right)^{-1}f^{\varepsilon}=0;roman_div roman_Δ ( italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 ;

For fixed ε𝜀\varepsilonitalic_ε, system (4.10) is equivalent with the diffusion problem

{div(με(x)uε)=fε(x)in Ωdivuε=0in Ωuε=0on Ω,casesdivsuperscript𝜇𝜀𝑥superscript𝑢𝜀superscript𝑓𝜀𝑥in Ωdivsuperscript𝑢𝜀0in Ωsuperscript𝑢𝜀0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu^{\varepsilon}\left(x% \right)\nabla u^{\varepsilon}\right)=f^{\varepsilon}\left(x\right)&\text{in }% \Omega\\ \operatorname{div}\,u^{\varepsilon}=0&\text{in }\Omega\\ u^{\varepsilon}=0&\text{on }\partial\Omega,\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ∇ italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) = italic_f start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω , end_CELL end_ROW end_ARRAY (4.11)

where divΔ(Lε)1fε=0.divΔsuperscriptsuperscript𝐿𝜀1superscript𝑓𝜀0\ \operatorname{div}\Delta\left(L^{\varepsilon}\right)^{-1}f^{\varepsilon}=0.roman_div roman_Δ ( italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 . Moreover, if fεfsuperscript𝑓𝜀𝑓f^{\varepsilon}\rightarrow fitalic_f start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT → italic_f in L2(Ω;n)superscript𝐿2Ωsuperscript𝑛L^{2}(\Omega;{\mathbb{R}}^{n})italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ), there is a function u0H01(Ω)superscript𝑢0subscriptsuperscript𝐻10Ωu^{0}\in H^{1}_{0}(\Omega)italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ) such that for ε0𝜀0\varepsilon\rightarrow 0italic_ε → 0 one has

uεu0weakly in H01(Ω;n)superscript𝑢𝜀superscript𝑢0weakly in subscriptsuperscript𝐻10Ωsuperscript𝑛u^{\varepsilon}\rightharpoonup u^{0}\quad\text{weakly in }H^{1}_{0}(\Omega;{% \mathbb{R}}^{n})italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ⇀ italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT weakly in italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) (4.12)

and u0superscript𝑢0u^{0}italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT is the unique solution of the homogenized problem

{div(Ahomu0)=fin Ωdivu0=0in Ωu0=0on Ω,casesdivsuperscript𝐴homsuperscript𝑢0𝑓in Ωdivsuperscript𝑢00in Ωsuperscript𝑢00on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(A^{\text{hom}}\,\nabla u^{0% }\right)=f&\text{in }\Omega\\ \operatorname{div}u^{0}=0&\text{in }\Omega\\ u^{0}=0&\text{on }\partial\Omega,\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT ∇ italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) = italic_f end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω , end_CELL end_ROW end_ARRAY (4.13)

where the matrix Ahomsuperscript𝐴homA^{\text{hom}}italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT is defined in (4.5).

For the case of an ε𝜀\varepsilonitalic_ε-dependent non reactive force fεsuperscript𝑓𝜀f^{\varepsilon}italic_f start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT strongly converging in L2(Ω;2)superscript𝐿2Ωsuperscript2L^{2}(\Omega;{\mathbb{R}}^{2})italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) to f𝑓fitalic_f, assuming that divΔ(Lε)1fε=0divΔsuperscriptsuperscript𝐿𝜀1superscript𝑓𝜀0\ \operatorname{div}\Delta\left(L^{\varepsilon}\right)^{-1}f^{\varepsilon}=0roman_div roman_Δ ( italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 and denoting hε:=Δ(Lε)1fε,assignsuperscript𝜀Δsuperscriptsuperscript𝐿𝜀1superscript𝑓𝜀\ h^{\varepsilon}:=-\Delta\left(L^{\varepsilon}\right)^{-1}f^{\varepsilon},\ italic_h start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT := - roman_Δ ( italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT , we may state the following localization result for problem (4.10), the analogue of Proposition 7.

Theorem 16

If there is an ε0>0subscript𝜀00\ \varepsilon_{0}>0italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 such that

0hε(x)RΓ for a.e. xΩ and0superscript𝜀𝑥𝑅Γ for a.e. 𝑥Ω and0\leq h^{\varepsilon}\left(x\right)\leq\frac{R}{\Gamma}\text{\ \ \ for a.e. }x\in\Omega\ \text{ and}0 ≤ italic_h start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ≤ divide start_ARG italic_R end_ARG start_ARG roman_Γ end_ARG for a.e. italic_x ∈ roman_Ω and (4.14)
rγhε(x) for a.e. xΩ,𝑟𝛾superscript𝜀𝑥 for a.e. 𝑥superscriptΩ\frac{r}{\gamma}\leq h^{\varepsilon}\left(x\right)\text{\ \ \ for a.e. }x\in% \Omega^{\prime},divide start_ARG italic_r end_ARG start_ARG italic_γ end_ARG ≤ italic_h start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , (4.15)

for every ε(0,ε0],𝜀0subscript𝜀0\ \varepsilon\in(0,\varepsilon_{0}],italic_ε ∈ ( 0 , italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] , then the velocity uεsuperscript𝑢𝜀\ u^{\varepsilon}italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT, unique solution of problem (4.11) satisfies

00\displaystyle 0 \displaystyle\leq uε(x)R for a.e. xΩ andformulae-sequencesuperscript𝑢𝜀𝑥𝑅 for a.e. 𝑥Ω and\displaystyle u^{\varepsilon}\left(x\right)\leq R\ \ \ \text{\ for a.e. }x\in% \Omega\text{\ \ and}italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ≤ italic_R for a.e. italic_x ∈ roman_Ω and
r𝑟\displaystyle ritalic_r \displaystyle\leq uε(x) for a.e. xΩ(0<εε0).superscript𝑢𝜀𝑥 for a.e. 𝑥superscriptΩ0𝜀subscript𝜀0\displaystyle u^{\varepsilon}\left(x\right)\text{\ \ \ for a.e. }x\in\Omega^{% \prime}\ \ \left(0<\varepsilon\leq\varepsilon_{0}\right).italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 < italic_ε ≤ italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) .

Convergence (4.12) holds and the homogenized velocity u0superscript𝑢0u^{0}italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, unique solution of problem (4.13) also satisfies

00\displaystyle 0 \displaystyle\leq u0(x)R for a.e. xΩ andformulae-sequencesuperscript𝑢0𝑥𝑅 for a.e. 𝑥Ω and\displaystyle u^{0}\left(x\right)\leq R\ \ \ \text{\ for a.e. }x\in\Omega\text% {\ \ and}italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_x ) ≤ italic_R for a.e. italic_x ∈ roman_Ω and
r𝑟\displaystyle ritalic_r \displaystyle\leq u0(x) for a.e. xΩ.superscript𝑢0𝑥 for a.e. 𝑥superscriptΩ\displaystyle u^{0}\left(x\right)\text{\ \ \ for a.e. }x\in\Omega^{\prime}\ \ .italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_x ) for a.e. italic_x ∈ roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

2b. For the analogue of the planar system (3.11) we consider, without loss of generality, the case f(x)=0𝑓𝑥0f(x)=0italic_f ( italic_x ) = 0

{div(με(x)uε)+pε=ϕε[uε](x)in Ωdivuε=0in Ωuε=0on Ω,casesdivsuperscript𝜇𝜀𝑥superscript𝑢𝜀superscript𝑝𝜀superscriptitalic-ϕ𝜀delimited-[]superscript𝑢𝜀𝑥in Ωdivsuperscript𝑢𝜀0in Ωsuperscript𝑢𝜀0on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(\mu^{\varepsilon}\left(x% \right)\nabla u^{\varepsilon}\right)+\nabla p^{\varepsilon}=\phi^{\varepsilon}% \left[u^{\varepsilon}\right]\left(x\right)&\text{in }\Omega\\ \operatorname{div}\,u^{\varepsilon}=0&\text{in }\Omega\\ u^{\varepsilon}=0&\text{on }\partial\Omega,\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_μ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( italic_x ) ∇ italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ) + ∇ italic_p start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = italic_ϕ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT [ italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ] ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω , end_CELL end_ROW end_ARRAY (4.16)

where the reaction force is of the form

ϕε[uε](x)=Lε(Δ)1(1Ωη(uε(.)))(x).\phi^{\varepsilon}\left[u^{\varepsilon}\right]\left(x\right)=L^{\varepsilon}% \left(-\Delta\right)^{-1}\left(1_{\Omega^{\prime}}\eta\left(u^{\varepsilon}% \left(.\right)\right)\right)\left(x\right).italic_ϕ start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT [ italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ] ( italic_x ) = italic_L start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_η ( italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ( . ) ) ) ( italic_x ) .

Using the notations from Section 3.2 and the assumptions (3.16) on the given function η,𝜂\ \eta,italic_η , we have the following result for the system (4.16), which is the analogue of Theorem 11.

Theorem 17

Assume that η𝜂\etaitalic_η is as in Theorem 11,

rmηγ,MηΓRformulae-sequence𝑟subscript𝑚𝜂𝛾subscript𝑀𝜂Γ𝑅r\leq m_{\eta}\gamma,\ \ \ M_{\eta}\Gamma\leq Ritalic_r ≤ italic_m start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT italic_γ , italic_M start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT roman_Γ ≤ italic_R (4.17)

and the spectral radius of the matrix \ \mathcal{M}caligraphic_M is strictly less than one. Then, there exists ε0>0subscript𝜀00\varepsilon_{0}>0italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0 such that for any ε<ε0𝜀subscript𝜀0\ \varepsilon<\varepsilon_{0}italic_ε < italic_ε start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, problem (4.16) has a unique solution (uε,0)superscript𝑢𝜀0\ \left(u^{\varepsilon},0\right)( italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT , 0 ) in DrRs×L02(Ω)superscriptsubscript𝐷𝑟𝑅𝑠subscriptsuperscript𝐿20ΩD_{rR}^{s}\times L^{2}_{0}\left(\Omega\right)italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT × italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ), where L02(Ω)subscriptsuperscript𝐿20ΩL^{2}_{0}\left(\Omega\right)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ) is the space of the L2(Ω)superscript𝐿2ΩL^{2}\left(\Omega\right)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) functions with mean value zero on ΩΩ\Omegaroman_Ω. Moreover, there is a function u1H01(Ω;2)superscript𝑢1subscriptsuperscript𝐻10Ωsuperscript2u^{1}\in H^{1}_{0}(\Omega;{\mathbb{R}}^{2})italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∈ italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) such that

uεu1weakly in H01(Ω;2),superscript𝑢𝜀superscript𝑢1weakly in subscriptsuperscript𝐻10Ωsuperscript2u^{\varepsilon}\rightharpoonup u^{1}\quad\text{weakly in }H^{1}_{0}(\Omega;{% \mathbb{R}}^{2}),italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT ⇀ italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT weakly in italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( roman_Ω ; blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ,

u1superscript𝑢1u^{1}italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT belongs to DrRssuperscriptsubscript𝐷𝑟𝑅𝑠D_{rR}^{s}italic_D start_POSTSUBSCRIPT italic_r italic_R end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT and it is the unique solution of the homogenized problem

{div(Ahomu1)=ϕ[u1](x)in Ωdivu1=0in Ωu1=0on Ω,casesdivsuperscript𝐴homsuperscript𝑢1italic-ϕdelimited-[]superscript𝑢1𝑥in Ωdivsuperscript𝑢10in Ωsuperscript𝑢10on Ω\left\{\begin{array}[]{ll}-\operatorname{div}\left(A^{\text{hom}}\,\nabla u^{1% }\right)=\phi[u^{1}](x)&\text{in }\Omega\\ \operatorname{div}u^{1}=0&\text{in }\Omega\\ u^{1}=0&\text{on }\partial\Omega,\end{array}\right.{ start_ARRAY start_ROW start_CELL - roman_div ( italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT ∇ italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ) = italic_ϕ [ italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ] ( italic_x ) end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL roman_div italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = 0 end_CELL start_CELL in roman_Ω end_CELL end_ROW start_ROW start_CELL italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT = 0 end_CELL start_CELL on ∂ roman_Ω , end_CELL end_ROW end_ARRAY (4.18)

where the matrix Ahomsuperscript𝐴homA^{\text{hom}}italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT is defined in (4.5) and ϕ[u1](x)=div(Ahom(Δ)1(1Ωη(u1(.)))(x))\phi[u_{1}](x)=-\operatorname{div}(A^{\text{hom}}\,\nabla(-\Delta)^{-1}(1_{% \Omega^{\prime}}\eta(u^{1}(.)))(x))italic_ϕ [ italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] ( italic_x ) = - roman_div ( italic_A start_POSTSUPERSCRIPT hom end_POSTSUPERSCRIPT ∇ ( - roman_Δ ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( 1 start_POSTSUBSCRIPT roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_η ( italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( . ) ) ) ( italic_x ) ).

So for both problems (4.10) and (4.16) we localize the velocity uε,superscript𝑢𝜀u^{\varepsilon},italic_u start_POSTSUPERSCRIPT italic_ε end_POSTSUPERSCRIPT , as far as the homogenized velocities u0superscript𝑢0u^{0}italic_u start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT and u1superscript𝑢1u^{1}italic_u start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT, respectively. For problem (4.16), multiple solutions can be obtained depending on the oscillatory properties of function η.𝜂\eta.italic_η .

5 Conclusions

We addressed in this paper the stationary Stokes system with variable viscosity and a reaction force term. We gave appropriate conditions on the reaction force term in order to obtain the localization and multiplicity of solutions. The method used combines concepts and results from the linear theory of PDEs and nonlinear functional analysis. In particular, we use the Moser-Harnack inequality, arguments of fixed point theory and Ekeland’s variational principle The results obtained apply to systems with strongly oscillating periodic viscosity and the corresponding homogenized systems. In this context, a significant gain is that our method makes possible the emergence of finitely or infinitely many solutions.

As a perspective, let us mention that the method can be applied to other classes of partial differential equations.

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