Penalization of history-dependent variational inequalities

Abstract

The present paper represents a continuation of Sofonea and Matei’s paper (Sofonea, M. and Matei, A. (2011) History-dependent quasivariational inequalities arising in contact mechanics.ย Eur. J. Appl. Math.ย 22, 471โ€“491). There a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact.

In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the material’s behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then we introduce the abstract variational inequalities together with their penalizations.

We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.

Authors

Mircea Sofonea
(Laboratoire de Mathรฉmatiques et Physique, Universitรฉ de Perpignan)

Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

history-dependent operator, variational inequality, penalization, viscoelastic material, frictionless contact, normal compliance, unilateral constraint, weak solution

City this paper as

M. Sofonea, F. Pฤƒtrulescu, Penalization of history-dependent variational inequalities, European J. Appl. Math., vol. 25, no. 2 (2014), pp. 155-176
DOI: 10.1017/S0956792513000363

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Cambridge University Press, Cambridge

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0956-7925

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1469-4425/e

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3177272

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1297.49016

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Penalization of History-Dependent Variational Inequalities

M. Sofonea 1, F. Pฤƒtrulescu 2
1 Laboratoire de Mathรฉmatiques et Physique
Universitรฉ de Perpignan, 52 Avenue de Paul Alduy, 66860 Perpignan, France
2 Tiberiu Popoviciu Institute of Numerical Analysis
P.O. Box 68-1, 400110 Cluj-Napoca, Romania
Abstract

The present paper represents a continuation of [21]. There, a new class of variational inequalities involving history-dependent operators was considered, an abstract existence and uniqueness result was proved and it was completed with a regularity result. Moreover, these results were used in the analysis of various frictional and frictionless models of contact. In this current paper we present a penalization method in the study of such inequalities. We start with an example which motivates our study; it concerns a mathematical model which describes the quasistatic contact between a viscoelastic body and a foundation; the materialโ€™s behaviour is modelled with a constitutive law with long memory, the contact is frictionless and is modelled with a multivalued normal compliance condition and unilateral constraint. Then, we introduce the abstract variational inequalities together with their penalizations. We prove the unique solvability of the penalized problems and the convergence of their solutions to the solution of the original problem, as the penalization parameter converges to zero. Finally, we turn back to our a contact model, apply our abstract results in the study of this problem and provide their mechanical interpretation.

2010 Mathematics Subject Classification : 49J40, 49J45, 47J20, 74M15, 74G25, 74G30.

Keywords: history-dependent operator, variational inequality, penalization, viscoelastic material, frictionless contact, normal compliance, unilateral constraint, weak solution.

1 Introduction

The theory of variational inequalities plays an important role in the study of nonlinear boundary value problems arising in mechanics, physics and engineering science. At the heart of this theory is the intrinsic inclusion of free boundaries in an elegant mathematical formulation. General results on the analysis of the variational inequalities, including existence and uniqueness results, can be found in [1, 2, 11, 13, 17, 22, for instance. Details concerning the numerical analysis of variational inequalities, including solution algorithms and error estimates, can be found in [6, 10]. References in the study of mathematical and numerical analysis of variational inequalities arising in hardening plasticity include [7, 8).

Phenomena of contact between deformable bodies abound in industry and everyday life. For this reason, considerable progress has been achieved recently in modelling, mathematical analysis and numerical simulations of various contact processes and, as a result, a general mathematical theory of contact mechanics is currently emerging. It is concerned with the mathematical structures which underlie general contact problems with different constitutive laws, i.e. materials, varied geometries and different contact conditions. To this end, it uses various mathematical concepts which include both variational and hemivariational inequalities and multivalued inclusions. An early attempt to study frictional contact problems within the framework of variational inequalities was made in [4]. An excellent reference on analysis and numerical approximations of contact problems involving elastic materials with or without friction is 12]. The variational analysis of various contact problems can be found in the monographs [5, 9, 10, 12, 16, 17, 20. The state of the art in the field can be found in the proceedings [14, 18, 24] and in the special issue [19], as well.

Existence, uniqueness and regularity results in the study of a new class of variational inequalities were proved in [21. There, the first trait of novelty lies in the fact that, unlike the results obtained in literature, the variational inequalities considered were defined on an unbounded interval of time. The second novelty was related to their special structure, which involves two nondifferentiable convex functionals, one of them depending on the history of the solution. This class of variational inequalities represents a general framework in which a large number of quasistatic contact problems, associated with various constitutive laws and frictional contact conditions, can be cast, as exemplified in [22].

Our intention in this current paper is to present a penalization method in the study of the variational inequalities introduced in [21 and to apply it to a new model of contact. Penalization methods in the study of elliptic variational inequalities were used by many authors, mainly for numerical reasons. Details can be found in [6] and the references therein. The main ingredient of these methods arises from the fact that they remove the constraints by considering penalized problems defined on the whole space; these approximative problems have unique solutions which converge to
the solutions of the original problems, as the penalization parameter converges to zero.

The rest of the paper is structured as follows. In Section 2 we present a new mathematical model of contact which is of applied interest and which motivates the abstract study we present in this paper. In Section 3 we state the abstract problem and recall its unique solvability, obtained in [21]. Then we state the penalized problems and prove our main result, Theorem 3.2. The proof of this theorem is given in Section 4. Further, we illustrate the use of the abstract results in the study of the contact model introduced in Section 2. To this end, in Section 5 we list the assumptions on the data and derive the variational formulation. Then we state and prove Theorem 5.1 which concerns the unique weak solvability of the model. Next, in Section 6, we use our abstract penalization method. Our main result in this section is given by Theorem 6.1 which states the existence of a unique weak solution of the penalized contact problems and its convergence to the weak solution of the original contact model. Finally, in Section 7, we present some concluding remarks.

2 A viscoelastic contact problem

The physical setting is as follows. A viscoelastic body occupies a bounded domain ฮฉโŠ‚โ„d(d=1,2,3)\Omega\subset\mathbb{R}^{d}(d=1,2,3) with a Lipschitz continuous boundary ฮ“\Gamma which is divided into three measurable parts ฮ“1,ฮ“2\Gamma_{1},\Gamma_{2} and ฮ“3\Gamma_{3}, such that meas (ฮ“1)>0\left(\Gamma_{1}\right)>0. The body is subject to the action of body forces of density ๐’‡0\boldsymbol{f}_{0}. We also assume that it is fixed on ฮ“1\Gamma_{1} and surface tractions of density ๐’‡2\boldsymbol{f}_{2} act on ฮ“2\Gamma_{2}. On ฮ“3\Gamma_{3}, the body is in frictionless contact with a deformable obstacle, the so-called foundation. We assume that the contact process is quasistatic, we study it in the interval of time โ„+=[0,โˆž)\mathbb{R}_{+}=[0,\infty), and we denote by ๐‚\boldsymbol{\nu} and ๐•Šd\mathbb{S}^{d} the outward unit normal at ฮ“\Gamma and the space of second order symmetric tensors on โ„d\mathbb{R}^{d}, respectively. Then, the classical formulation of the contact problem we consider in the rest of this paper is the following.

Problem ๐’ฌ\mathcal{Q}. Find a displacement field ๐’–:ฮฉร—โ„+โ†’โ„d\boldsymbol{u}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d} and a stress field ๐ˆ\boldsymbol{\sigma} : ฮฉร—โ„+โ†’๐•Šd\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d} such that

๐ˆ(t)=๐’œ๐œบ(๐’–(t))+โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–(s))๐‘‘s\displaystyle\boldsymbol{\sigma}(t)=\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t))+\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}(\boldsymbol{u}(s))ds in ฮฉ\displaystyle\text{ in }\quad\Omega (2.1)
Divโก๐ˆ(t)+๐’‡0(t)=๐ŸŽ\displaystyle\operatorname{Div}\boldsymbol{\sigma}(t)+\boldsymbol{f}_{0}(t)=\mathbf{0} in ฮฉ\displaystyle\text{ in }\quad\Omega (2.2)
๐’–(t)=๐ŸŽ\displaystyle\boldsymbol{u}(t)=\mathbf{0} on ฮ“1\displaystyle\text{ on }\quad\Gamma_{1} (2.3)
๐ˆ(t)๐‚=๐’‡2(t)\displaystyle\boldsymbol{\sigma}(t)\boldsymbol{\nu}=\boldsymbol{f}_{2}(t) on ฮ“2,\displaystyle\text{ on }\quad\Gamma_{2}, (2.4)
๐ˆฯ„(t)=๐ŸŽ\displaystyle\boldsymbol{\sigma}_{\tau}(t)=\mathbf{0} on ฮ“3,\displaystyle\text{ on }\quad\Gamma_{3}, (2.5)

for all tโˆˆโ„+t\in\mathbb{R}_{+}, and there exists ฮพ:ฮฉร—โ„+โ†’โ„\xi:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R} which satisfies

uฮฝ(t)โ‰คg,ฯƒฮฝ(t)+p(uฮฝ(t))+ฮพ(t)โ‰ค0(uฮฝ(t)โˆ’g)(ฯƒฮฝ(t)+p(uฮฝ(t))+ฮพ(t))=00โ‰คฮพ(t)โ‰คFฮพ(t)=0 if uฮฝ(t)<0ฮพ(t)=F if uฮฝ(t)>0} on ฮ“3\left.\begin{array}[]{l}u_{\nu}(t)\leq g,\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t)\leq 0\\ \left(u_{\nu}(t)-g\right)\left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t)\right)=0\\ 0\leq\xi(t)\leq F\\ \xi(t)=0\text{ if }u_{\nu}(t)<0\\ \xi(t)=F\text{ if }u_{\nu}(t)>0\end{array}\right\}\quad\text{ on }\quad\Gamma_{3}

for all tโˆˆโ„+t\in\mathbb{R}_{+}.
Here and below, in order to simplify the notation, we do not indicate explicitly the dependence of various functions on the spatial variable ๐’™\boldsymbol{x}. Equation (2.1) represents the viscoelastic constitutive law with long memory in which ๐’œ\mathcal{A} is the elasticity operator, โ„ฌ\mathcal{B} represents the relaxation tensor and ฮต(๐’–)\varepsilon(\boldsymbol{u}) denotes the linearized strain tensor. Equation (2.2) represents the equation of equilibrium in which Div denotes the divergence operator for tensor valued functions. Conditions (2.3) and (2.4) are the displacement boundary condition and the traction boundary condition, respectively. Condition (2.5) is the frictionless condition and it shows that the tangential stress on the contact surface, denoted by ๐ˆฯ„\boldsymbol{\sigma}_{\tau}, vanishes. More details on the equations and conditions (2.1)-(2.5) can be found in 22.

We now describe the contact condition (2.6) in which our main interest lies and which represents the main novelty of the model. Here ฯƒฮฝ\sigma_{\nu} denotes the normal stress, uฮฝu_{\nu} is the normal displacement and uฮฝ+u_{\nu}^{+}may be interpreted as the penetration of the bodyโ€™s surface asperities and those of the foundation. Moreover, pp is a Lipschitz continuous increasing function which vanishes for a negative argument, FF is a positive function and g>0g>0. This condition can be derived in the following way. Let tโˆˆโ„+t\in\mathbb{R}_{+}be given. First, we assume that the penetration is limited by the bound gg and, therefore, the normal displacement satisfies the inequality

uฮฝ(t)โ‰คg on ฮ“3.u_{\nu}(t)\leq g\quad\text{ on }\Gamma_{3}. (2.7)

Next, we assume that the normal stress has an additive decomposition of the form

ฯƒฮฝ(t)=ฯƒฮฝD(t)+ฯƒฮฝR(t)+ฯƒฮฝM(t) on ฮ“3\sigma_{\nu}(t)=\sigma_{\nu}^{D}(t)+\sigma_{\nu}^{R}(t)+\sigma_{\nu}^{M}(t)\quad\text{ on }\Gamma_{3} (2.8)

in which the function ฯƒฮฝD(t)\sigma_{\nu}^{D}(t) describes the deformability of the foundation and the functions ฯƒฮฝR(t),ฯƒฮฝM(t)\sigma_{\nu}^{R}(t),\sigma_{\nu}^{M}(t) describe the rigidity and the memory properties of the foundation, respectively. We assume that ฯƒฮฝD(t)\sigma_{\nu}^{D}(t) satisfies a normal compliance contact condition, that is

โˆ’ฯƒฮฝD(t)=p(uฮฝ(t)) on ฮ“3.-\sigma_{\nu}^{D}(t)=p\left(u_{\nu}(t)\right)\quad\text{ on }\Gamma_{3}. (2.9)

The part ฯƒฮฝR(t)\sigma_{\nu}^{R}(t) of the normal stress satisfies the Signorini condition in the form with a gap function, i.e.

ฯƒฮฝR(t)โ‰ค0,ฯƒฮฝR(t)(uฮฝ(t)โˆ’g)=0 on ฮ“3.\sigma_{\nu}^{R}(t)\leq 0,\quad\sigma_{\nu}^{R}(t)\left(u_{\nu}(t)-g\right)=0\quad\text{ on }\Gamma_{3}. (2.10)

Finally, the function ฯƒฮฝM(t)\sigma_{\nu}^{M}(t) satisfies the condition

{|ฯƒฮฝM(t)|โ‰คF,ฯƒฮฝM(t)=0 if uฮฝ(t)<0,โˆ’ฯƒฮฝM(t)=F if uฮฝ(t)>0 on ฮ“3\left\{\begin{array}[]{l}\left|\sigma_{\nu}^{M}(t)\right|\leq F,\quad\sigma_{\nu}^{M}(t)=0\quad\text{ if }\quad u_{\nu}(t)<0,\\ -\sigma_{\nu}^{M}(t)=F\quad\text{ if }\quad u_{\nu}(t)>0\end{array}\quad\text{ on }\Gamma_{3}\right.

We combine (2.8), (2.9) and write โˆ’ฯƒฮฝM(t)=ฮพ(t)-\sigma_{\nu}^{M}(t)=\xi(t) to see that

ฯƒฮฝR(t)=ฯƒฮฝ(t)+p(uฮฝ(t))+ฮพ(t) on ฮ“3.\sigma_{\nu}^{R}(t)=\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t)\quad\text{ on }\Gamma_{3}. (2.12)

Then we substitute equality (2.12) in (2.10) and use (2.7), (2.11) to obtain the contact condition (2.6).

We now present additional details of the contact condition (2.6). The inequalities and equalities below in this section are valid at an arbitrary point ๐’™โˆˆฮ“3\boldsymbol{x}\in\Gamma_{3}. First, we recall that (2.6) describes a condition with unilateral constraint, since inequality (2.7) holds at each moment of time. Next, assume that at a given moment tt there is separation between the body and the foundation, i.e. uฮฝ(t)<0u_{\nu}(t)<0. Then, since p(uฮฝ(t))=0p\left(u_{\nu}(t)\right)=0, (2.6) shows that ฯƒฮฝ(t)=0\sigma_{\nu}(t)=0, i.e. the reaction of the foundation vanishes. Note that the same behaviour of the normal stress is described both in the classical normal compliance condition and in the Signorini contact condition, when there is separation. Assume now that at the moment tt there is penetration which did not reach the bound gg, i.e. 0<uฮฝ(t)<g0<u_{\nu}(t)<g. Then (2.6) yields

โˆ’ฯƒฮฝ(t)=p(uฮฝ(t))+F-\sigma_{\nu}(t)=p\left(u_{\nu}(t)\right)+F (2.13)

This equality shows that, at the moment tt, the reaction of the foundation depends on the penetration and represents a normal compliance-type condition. Note that (2.6) also shows that if at the moment tt we have penetration which satisfies 0<uฮฝ(t)<g0<u_{\nu}(t)<g then โˆ’ฯƒฮฝ(t)โ‰ฅF-\sigma_{\nu}(t)\geq F. Indeed, if 0<uฮฝ(t)<g0<u_{\nu}(t)<g then (2.13) holds and this implies that โˆ’ฯƒฮฝ(t)โ‰ฅF-\sigma_{\nu}(t)\geq F. We conclude from above that if โˆ’ฯƒฮฝ(t)<F-\sigma_{\nu}(t)<F then there is no penetration and, therefore, FF represents a yield limit of the normal pressure, under which the penetration is not possible. This kind of behaviour characterizes a rigidelastic foundation.

In conclusion, condition (2.6) shows that when there is separation between the bodyโ€™s surface and the foundation then the normal stress vanishes; the penetration arises only if the normal stress reaches the critical value FF; when there is penetration the contact follows a normal compliance condition of the form (2.13) but up to the limit gg and then, when this limit is reached, the contact follows a Signorini-type unilateral condition with the gap gg. For this reason we refer to this condition as to a multivalued normal compliance contact condition with unilateral constraint. It can be interpreted physically as follows. The foundation is assumed to be made of a hard material covered by a thin layer of a soft material with thickness gg. The soft material has a rigid-elastic behaviour, i.e. is deformable, allows penetration, but only if the normal stress arrives to the yield value FF; the contact with this layer is modelled
with normal compliance, as shown in equality (2.13). The hard material is perfectly rigid and, therefore, it does not allow penetration; the contact with this material is modelled with the Signorini contact condition.

Two questions arise in the study of the unilateral contact problem ๐’ฌ\mathcal{Q}. The first one concerns its unique solvability; the second one concerns the approach of the solution by the solution of a contact model with normal compliance without unilateral constraint. The answers to the questions above are provided by the variational analysis of this contact problem, presented in Section 5 and 6. This analysis is carried out based on the abstract existence, uniqueness and convergence result that we present in the next section.

3 Abstract problem and main result

Everywhere below we use the notation โ„•โˆ—\mathbb{N}^{*} for the set of positive integers and โ„+=[0,โˆž)\mathbb{R}_{+}=[0,\infty). For each normed space XX we use the notation C(โ„+;X)C\left(\mathbb{R}_{+};X\right) for the space of continuous functions defined on โ„+\mathbb{R}_{+}with values in XX. For a subset KโŠ‚XK\subset X we still use the symbol C(โ„+;K)C\left(\mathbb{R}_{+};K\right) for the set of continuous functions defined on โ„+\mathbb{R}_{+}with values in KK. It is well known that, if XX is a Banach space, then C(โ„+;X)C\left(\mathbb{R}_{+};X\right) can be organized in a canonical way as a Frรฉchet space, i.e. as a complete metric space in which the corresponding topology is induced by a countable family of seminorms. Details can be found in 3 and [15], for instance. Here we only need to recall that the convergence of a sequence (xk)k\left(x_{k}\right)_{k} to the element xx, in the space C(โ„+;X)C\left(\mathbb{R}_{+};X\right), can be described as follows:

{xkโ†’x in C(โ„+;X) as kโ†’โˆž if and only if maxrโˆˆ[0,n]โกโ€–xk(r)โˆ’x(r)โ€–Xโ†’0 as kโ†’โˆž, for all nโˆˆโ„•โˆ—.\left\{\begin{array}[]{l}x_{k}\rightarrow x\quad\text{ in }C\left(\mathbb{R}_{+};X\right)\text{ as }k\rightarrow\infty\text{ if and only if }\\ \max_{r\in[0,n]}\left\|x_{k}(r)-x(r)\right\|_{X}\rightarrow 0\text{ as }k\rightarrow\infty,\text{ for all }n\in\mathbb{N}^{*}.\end{array}\right.

Consider now a real Hilbert space XX with inner product (โ‹…,โ‹…)X(\cdot,\cdot)_{X} and associated norm โˆฅโ‹…โˆฅX\|\cdot\|_{X}. Also, let KK be a subset of XX, let A:Xโ†’X,๐’ฎ:C(โ„+;X)โ†’C(โ„+;X)A:X\rightarrow X,\mathcal{S}:C\left(\mathbb{R}_{+};X\right)\rightarrow C\left(\mathbb{R}_{+};X\right) be two operators, and let j:Xโ†’โ„,f:โ„+โ†’Xj:X\rightarrow\mathbb{R},f:\mathbb{R}_{+}\rightarrow X be two functions. We assume in what follows that

K is a nonempty closed convex subset of X,K\text{ is a nonempty closed convex subset of }X, (3.2)

and AA is strongly monotone and Lipschitz continuous operator, i.e.

{ (a) There exists m>0 such that (Au1โˆ’Au2,u1โˆ’u2)Xโ‰ฅmโ€–u1โˆ’u2โ€–X2โˆ€u1,u2โˆˆX. (b) There exists M>0 such that โ€–Au1โˆ’Au2โ€–Xโ‰คMโ€–u1โˆ’u2โ€–Xโˆ€u1,u2โˆˆX.\left\{\begin{array}[]{l}\text{ (a) There exists }m>0\text{ such that }\\ \quad\left(Au_{1}-Au_{2},u_{1}-u_{2}\right)_{X}\geq m\left\|u_{1}-u_{2}\right\|_{X}^{2}\\ \quad\forall u_{1},u_{2}\in X.\\ \text{ (b) There exists }M>0\text{ such that }\\ \left\|Au_{1}-Au_{2}\right\|_{X}\leq M\left\|u_{1}-u_{2}\right\|_{X}\quad\forall u_{1},u_{2}\in X.\end{array}\right.

Moreover, we assume that the operator ๐’ฎ\mathcal{S} satisfies the following condition:

{ For every nโˆˆโ„•โˆ— there exists dn>0 such that โ€–๐’ฎu1(t)โˆ’๐’ฎu2(t)โ€–Xโ‰คdnโˆซ0tโ€–u1(s)โˆ’u2(s)โ€–X๐‘‘sโˆ€u1,u2โˆˆC(โ„+;X),โˆ€tโˆˆ[0,n]\left\{\begin{array}[]{l}\text{ For every }n\in\mathbb{N}^{*}\text{ there exists }d_{n}>0\text{ such that }\\ \left\|\mathcal{S}u_{1}(t)-\mathcal{S}u_{2}(t)\right\|_{X}\leq d_{n}\int_{0}^{t}\left\|u_{1}(s)-u_{2}(s)\right\|_{X}ds\\ \forall u_{1},u_{2}\in C\left(\mathbb{R}_{+};X\right),\forall t\in[0,n]\end{array}\right.

Following the terminology in [21, 22] we refer to an operator ๐’ฎ\mathcal{S} which satisfies (3.4) as a history-dependent operator. Finally, we suppose that

j:Xโ†’โ„ is a proper convex lower semicontinuous function.\displaystyle j:X\rightarrow\mathbb{R}\text{ is a proper convex lower semicontinuous function. } (3.5)
fโˆˆC(โ„+;X).\displaystyle f\in C\left(\mathbb{R}_{+};X\right)\text{. } (3.6)

With the data above, we consider the following problem.
Problem ๐’ซ\mathcal{P}. Find a function u:โ„+โ†’Xu:\mathbb{R}_{+}\rightarrow X such that, for all tโˆˆโ„+t\in\mathbb{R}_{+}, the inequality below holds:

u(t)\displaystyle u(t) โˆˆK,(Au(t),vโˆ’u(t))X+(๐’ฎu(t),vโˆ’u(t))X\displaystyle\in K,\quad(Au(t),v-u(t))_{X}+(\mathcal{S}u(t),v-u(t))_{X} (3.7)
+j(v)โˆ’j(u(t))โ‰ฅ(f(t),vโˆ’u(t))Xโˆ€vโˆˆK\displaystyle+j(v)-j(u(t))\geq(f(t),v-u(t))_{X}\quad\forall v\in K

Following [21, 22] we refer to (3.7) as a history-dependent variational inequality. It represents the framework in which the variational formulation of a large number of contact problems can be cast, with the appropriate choice of spaces and operators. Details can be found in [9, 21, 22, 23] and the references therein. The solvability of Problem ๐’ซ\mathcal{P} is provided by the following existence and uniqueness result, proved in [21.

Theorem 3.1 Let XX be a Hilbert space and assume that (3.2)-(3.6) hold. Then, Problem ๐’ซ\mathcal{P} has a unique solution uโˆˆC(โ„+;K)u\in C\left(\mathbb{R}_{+};K\right).

In order to formulate the penalized problems associated to Problem ๐’ซ\mathcal{P} we consider an operator G:Xโ†’XG:X\rightarrow X which satisfies the following conditions:

{ (a) (Guโˆ’Gv,uโˆ’v)Xโ‰ฅ0โˆ€u,vโˆˆX. (b) There exists L>0 such that โ€–Guโˆ’Gvโ€–Xโ‰คLโ€–uโˆ’vโ€–Xโˆ€u,vโˆˆX. (c) (Gu,vโˆ’u)Xโ‰ค0โˆ€uโˆˆX,vโˆˆK. (d) Gu=0X iff uโˆˆK.\left\{\begin{array}[]{l}\text{ (a) }(Gu-Gv,u-v)_{X}\geq 0\quad\forall u,v\in X.\\ \text{ (b) There exists }L>0\text{ such that }\\ \quad\|Gu-Gv\|_{X}\leq L\|u-v\|_{X}\quad\forall u,v\in X.\\ \text{ (c) }(Gu,v-u)_{X}\leq 0\quad\forall u\in X,v\in K.\\ \text{ (d) }Gu=0_{X}\text{ iff }u\in K.\end{array}\right.

Note that conditions (3.8) (a) and (b) show that GG is a monotone Lipschtz continuous operator. Also, note that such an operator GG always exists. For example consider the operator G:Xโ†’XG:X\rightarrow X defined by

Gu=uโˆ’PKu,โˆ€uโˆˆK,Gu=u-P_{K}u,\quad\forall u\in K,

where PK:Xโ†’KP_{K}:X\rightarrow K represents the projection operator onto KK. Then, using the properties of the projections, it is easy to see that the operator GG satisfies condition (3.8).

Next, for each ฮผ>0\mu>0 we consider the following problem.
Problem ๐’ซฮผ\mathcal{P}_{\mu}. Find a function uฮผ:โ„+โ†’Xu_{\mu}:\mathbb{R}_{+}\rightarrow X such that, for all tโˆˆโ„+t\in\mathbb{R}_{+}, the inequality below holds:

(Auฮผ(t)\displaystyle\left(Au_{\mu}(t)\right. ,vโˆ’uฮผ(t))X+(๐’ฎuฮผ(t),vโˆ’uฮผ(t))X+1ฮผ(Guฮผ(t),vโˆ’uฮผ(t))X\displaystyle\left.,v-u_{\mu}(t)\right)_{X}+\left(\mathcal{S}u_{\mu}(t),v-u_{\mu}(t)\right)_{X}+\frac{1}{\mu}\left(Gu_{\mu}(t),v-u_{\mu}(t)\right)_{X} (3.9)
+j(v)โˆ’j(uฮผ(t))โ‰ฅ(f(t),vโˆ’uฮผ(t))X\displaystyle+j(v)-j\left(u_{\mu}(t)\right)\geq\left(f(t),v-u_{\mu}(t)\right)_{X} โˆ€vโˆˆX\displaystyle\forall v\in X

Note that, in contrast to Problem ๐’ซ\mathcal{P}, in Problem ๐’ซฮผ\mathcal{P}_{\mu} the constraint u(t)โˆˆKu(t)\in K is removed and is replaced with an additional term which contains the penalization parameter ฮผ\mu. For this reason, we refer to Problem ๐’ซฮผ\mathcal{P}_{\mu} as a penalized problem associated to Problem ๐’ซ\mathcal{P}.

We have the following existence, uniqueness and convergence result, which represents the main result of this section.

Theorem 3.2 Let XX be a Hilbert space and assume that (3.2)-(3.6), (3.8) hold. Then:

  1. 1.

    For each ฮผ>0\mu>0 Problem ๐’ซฮผ\mathcal{P}_{\mu} has a unique solution which satisfies uฮผโˆˆC(โ„+;X)u_{\mu}\in C\left(\mathbb{R}_{+};X\right).

  2. 2.

    The solution uฮผu_{\mu} of Problem ๐’ซฮผ\mathcal{P}_{\mu} converges to the solution uu of Problem ๐’ซ\mathcal{P}, that is

โ€–uฮผ(t)โˆ’u(t)โ€–Xโ†’0 as ฮผโ†’0\left\|u_{\mu}(t)-u(t)\right\|_{X}\rightarrow 0\quad\text{ as }\quad\mu\rightarrow 0 (3.10)

for each tโˆˆโ„+t\in\mathbb{R}_{+}.
Note that the convergence (3.10) above is understood in the following sense: for each tโˆˆโ„+t\in\mathbb{R}_{+}and for every sequence {ฮผn}โŠ‚โ„+\left\{\mu_{n}\right\}\subset\mathbb{R}_{+}converging to 0 as nโ†’โˆžn\rightarrow\infty we have uฮผn(t)โ†’u(t)u_{\mu_{n}}(t)\rightarrow u(t) as nโ†’โˆžn\rightarrow\infty.

4 Proof of Theorem 3.2

The proof of Theorem 3.2 will be carried out in several steps that we present in what follows. To this end, below in this section we assume that (3.2)-(3.6), (3.8) hold and
we denote by cc a positive constant which may depend on t,A,๐’ฎ,j,ft,A,\mathcal{S},j,f and uu, but is independent of ฮผ\mu, and whose value may change from line to line. The following lemma shows the unique solvability of the nonlinear inequality (3.9).

Lemma 4.1 For each ฮผ>0\mu>0 there exists a unique function uฮผโˆˆC(โ„+;X)u_{\mu}\in C\left(\mathbb{R}_{+};X\right) which satisfies the inequality (3.9) for all tโˆˆโ„+t\in\mathbb{R}_{+}.

Proof. Let ฮผ>0\mu>0. Using (3.3) and (3.8) it is easy to show that the operator

vโŸผAv+1ฮผGvv\longmapsto Av+\frac{1}{\mu}Gv

is a strongly monotone Lipschitz continuous operator on XX. Lemma 4.1 is now a consequence of Theorem 3.1 used with K=XK=X.

Next, we consider the following intermediate problem.
Problem ๐’ซ~ฮผ\widetilde{\mathcal{P}}_{\mu}. Find a function u~ฮผ:โ„+โ†’X\widetilde{u}_{\mu}:\mathbb{R}_{+}\rightarrow X such that, for all tโˆˆโ„+t\in\mathbb{R}_{+}, the inequality below holds:

(Au~ฮผ(t),vโˆ’u~ฮผ(t))X+(๐’ฎu(t),vโˆ’u~ฮผ(t))X+1ฮผ(Gu~ฮผ(t),vโˆ’u~ฮผ(t))X\displaystyle\left(A\widetilde{u}_{\mu}(t),v-\widetilde{u}_{\mu}(t)\right)_{X}+\left(\mathcal{S}u(t),v-\widetilde{u}_{\mu}(t)\right)_{X}+\frac{1}{\mu}\left(G\widetilde{u}_{\mu}(t),v-\widetilde{u}_{\mu}(t)\right)_{X} (4.1)
+j(v)โˆ’j(u~ฮผ(t))โ‰ฅ(f(t),vโˆ’u~ฮผ(t))Xโˆ€vโˆˆX\displaystyle+j(v)-j\left(\widetilde{u}_{\mu}(t)\right)\geq\left(f(t),v-\widetilde{u}_{\mu}(t)\right)_{X}\quad\forall v\in X

Note that inequality (3.9) is a history-dependent variational inequality, since the operator ๐’ฎ\mathcal{S} is applied to the unknown uฮผu_{\mu}. In contrast, the variational inequality (4.1) is a time-dependent variational inequality, since here ๐’ฎu\mathcal{S}u is a given function. The following lemma shows the unique solvability of the nonlinear inequality (4.1).

Lemma 4.2 For each ฮผ>0\mu>0 there exists a unique function u~ฮผโˆˆC(โ„+;X)\widetilde{u}_{\mu}\in C\left(\mathbb{R}_{+};X\right) which satisfies the inequality (4.1), for all tโˆˆโ„+t\in\mathbb{R}_{+}.

Proof. The proof is obtained by similar arguments to those used in the proof of Lemma 4.1.

Next we investigate the properties of the sequence {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\} for a fixed tโˆˆโ„+t\in\mathbb{R}_{+}.

Lemma 4.3 For each tโˆˆโ„+t\in\mathbb{R}_{+}there exists a subsequence of the sequence {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\}, again denoted {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\}, which converges weakly to u(t)u(t), i.e.

u~ฮผ(t)โ‡€u(t) in X as ฮผโ†’0.\widetilde{u}_{\mu}(t)\rightharpoonup u(t)\quad\text{ in }X\quad\text{ as }\mu\rightarrow 0. (4.2)

Proof. Let tโˆˆโ„+,ฮผ>0t\in\mathbb{R}_{+},\mu>0 and let v0โˆˆKv_{0}\in K. We use (4.1) to obtain

(Au~ฮผ(t),v0โˆ’u~ฮผ(t))X+(๐’ฎu(t),v0โˆ’u~ฮผ(t))X+1ฮผ(Gu~ฮผ(t),v0โˆ’u~ฮผ(t))X+j(v0)โˆ’j(u~ฮผ(t))โ‰ฅ(f(t),v0โˆ’u~ฮผ(t))X\begin{gathered}\left(A\widetilde{u}_{\mu}(t),v_{0}-\widetilde{u}_{\mu}(t)\right)_{X}+\left(\mathcal{S}u(t),v_{0}-\widetilde{u}_{\mu}(t)\right)_{X}+\frac{1}{\mu}\left(G\widetilde{u}_{\mu}(t),v_{0}-\widetilde{u}_{\mu}(t)\right)_{X}\\ +j\left(v_{0}\right)-j\left(\widetilde{u}_{\mu}(t)\right)\geq\left(f(t),v_{0}-\widetilde{u}_{\mu}(t)\right)_{X}\end{gathered}

and, therefore,

(Au~ฮผ(t)โˆ’Av0,u~ฮผ(t)โˆ’v0)Xโ‰ค(Av0,v0โˆ’u~ฮผ(t))X\displaystyle\left(A\widetilde{u}_{\mu}(t)-\right.\left.Av_{0},\widetilde{u}_{\mu}(t)-v_{0}\right)_{X}\leq\left(Av_{0},v_{0}-\widetilde{u}_{\mu}(t)\right)_{X} (4.3)
+(๐’ฎu(t),v0โˆ’u~ฮผ(t))X+1ฮผ(Gu~ฮผ(t),v0โˆ’u~ฮผ(t))X\displaystyle+\left(\mathcal{S}u(t),v_{0}-\widetilde{u}_{\mu}(t)\right)_{X}+\frac{1}{\mu}\left(G\widetilde{u}_{\mu}(t),v_{0}-\widetilde{u}_{\mu}(t)\right)_{X}
+j(v0)โˆ’j(u~ฮผ(t))+(f(t),u~ฮผ(t)โˆ’v0)X\displaystyle+j\left(v_{0}\right)-j\left(\widetilde{u}_{\mu}(t)\right)+\left(f(t),\widetilde{u}_{\mu}(t)-v_{0}\right)_{X}

We use (3.5) to see that there exist ฯ‰โˆˆX\omega\in X and ฮฑโˆˆโ„\alpha\in\mathbb{R}, which do not depend on tt, such that

j(v)โ‰ฅ(ฯ‰,v)X+ฮฑโˆ€vโˆˆVj(v)\geq(\omega,v)_{X}+\alpha\quad\forall v\in V

and, therefore,

j(u~ฮผ(t))โ‰ฅ(ฯ‰,u~ฮผ(t))X+ฮฑ.j\left(\widetilde{u}_{\mu}(t)\right)\geq\left(\omega,\widetilde{u}_{\mu}(t)\right)_{X}+\alpha. (4.4)

Then, we combine (4.3), (3.3), (3.8) (c) and (4.4) to find that

mโ€–u~ฮผ(t)โˆ’v0โ€–X2\displaystyle m\left\|\widetilde{u}_{\mu}(t)-v_{0}\right\|_{X}^{2} (4.5)
โ‰ค(โ€–Av0โ€–X+โ€–๐’ฎu(t)โ€–X+โ€–f(t)โ€–X+โ€–ฯ‰โ€–X)โ€–u~ฮผ(t)โˆ’v0โ€–X\displaystyle\quad\leq\left(\left\|Av_{0}\right\|_{X}+\|\mathcal{S}u(t)\|_{X}+\|f(t)\|_{X}+\|\omega\|_{X}\right)\left\|\widetilde{u}_{\mu}(t)-v_{0}\right\|_{X}
+|j(v0)|+|ฮฑ|+โˆฅฯ‰โˆฅโˆฅโˆฅXv0โˆฅX\displaystyle\quad+\left|j\left(v_{0}\right)\right|+|\alpha|+\|\omega\|\left\|{}_{X}\right\|v_{0}\|_{X}

We use now (4.5), the elementary inequality

x,a,bโ‰ฅ0 and x2โ‰คax+bโŸนx2โ‰คa2+2bx,a,b\geq 0\quad\text{ and }\quad x^{2}\leq ax+b\Longrightarrow x^{2}\leq a^{2}+2b

and the triangle inequality

โ€–u~ฮผ(t)โ€–Xโ‰คโ€–u~ฮผ(t)โˆ’v0โ€–X+โ€–v0โ€–X.\left\|\widetilde{u}_{\mu}(t)\right\|_{X}\leq\left\|\widetilde{u}_{\mu}(t)-v_{0}\right\|_{X}+\left\|v_{0}\right\|_{X}.

As a result we deduce that there exists c>0c>0 which depends on v0v_{0} but does not depend on ฮผ\mu such that

โ€–u~ฮผ(t)โ€–Xโ‰คc.\left\|\widetilde{u}_{\mu}(t)\right\|_{X}\leq c. (4.6)

Inequality (4.6) shows that the sequence {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\} is bounded in XX. Therefore, it follows that there exists a subsequence of the sequence {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\}, again denoted {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\} and an element u~(t)โˆˆX\widetilde{u}(t)\in X such that

u~ฮผ(t)โ‡€u~(t) in X as ฮผโ†’0.\widetilde{u}_{\mu}(t)\rightharpoonup\widetilde{u}(t)\quad\text{ in }\quad X\quad\text{ as }\mu\rightarrow 0. (4.7)

Next, we investigate the properties of the element u~(t)โˆˆX\widetilde{u}(t)\in X. First of all, we show that u~(t)โˆˆK\widetilde{u}(t)\in K. To this end, we use (4.1) to deduce that

1ฮผ(Gu~ฮผ(t),u~ฮผ(t)โˆ’v)Xโ‰ค(Au~ฮผ(t),vโˆ’u~ฮผ(t))X+(๐’ฎu(t),vโˆ’u~ฮผ(t))X\displaystyle\frac{1}{\mu}\left(G\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-v\right)_{X}\leq\left(A\widetilde{u}_{\mu}(t),v-\widetilde{u}_{\mu}(t)\right)_{X}+\left(\mathcal{S}u(t),v-\widetilde{u}_{\mu}(t)\right)_{X} (4.8)
+j(v)โˆ’j(u~ฮผ(t))+(f(t),u~ฮผ(t)โˆ’v)Xโˆ€vโˆˆX\displaystyle+j(v)-j\left(\widetilde{u}_{\mu}(t)\right)+\left(f(t),\widetilde{u}_{\mu}(t)-v\right)_{X}\quad\forall v\in X

We now write

Au~ฮผ(t)=Au~ฮผ(t)โˆ’A0X+A0XA\widetilde{u}_{\mu}(t)=A\widetilde{u}_{\mu}(t)-A0_{X}+A0_{X}

then we use the Lipschitz continuity of the operator AA and inequality (4.4) to obtain that

1ฮผ(Gu~ฮผ(t),u~ฮผ(t)โˆ’v)Xโ‰ค(Au~ฮผ(t)โˆ’A0X,vโˆ’u~ฮผ(t))X+(A0X,vโˆ’u~ฮผ(t))X\displaystyle\frac{1}{\mu}\left(G\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-v\right)_{X}\leq\left(A\widetilde{u}_{\mu}(t)-A0_{X},v-\widetilde{u}_{\mu}(t)\right)_{X}+\left(A0_{X},v-\widetilde{u}_{\mu}(t)\right)_{X}
+(๐’ฎu(t),vโˆ’u~ฮผ(t))X+j(v)โˆ’j(u~ฮผ(t))+(f(t),u~ฮผ(t)โˆ’v)X\displaystyle\quad+\left(\mathcal{S}u(t),v-\widetilde{u}_{\mu}(t)\right)_{X}+j(v)-j\left(\widetilde{u}_{\mu}(t)\right)+\left(f(t),\widetilde{u}_{\mu}(t)-v\right)_{X}
โ‰ค(Mโ€–u~ฮผ(t)โ€–X+โ€–A0Xโ€–X+โ€–๐’ฎu(t)โ€–X+โ€–f(t)โ€–X)(โ€–vโ€–X+โ€–u~ฮผ(t)โ€–X)\displaystyle\leq\left(M\left\|\widetilde{u}_{\mu}(t)\right\|_{X}+\left\|A0_{X}\right\|_{X}+\|\mathcal{S}u(t)\|_{X}+\|f(t)\|_{X}\right)\left(\|v\|_{X}+\left\|\widetilde{u}_{\mu}(t)\right\|_{X}\right)
+|j(v)|+โ€–u~ฮผ(t)โ€–Xโ€–ฯ‰โ€–X+|ฮฑ|\displaystyle\quad+|j(v)|+\left\|\widetilde{u}_{\mu}(t)\right\|_{X}\|\omega\|_{X}+|\alpha|

We combine now this inequality and (4.6) to see that there exists a positive constant cc which depends on t,A,f,๐’ฎ,j,ut,A,f,\mathcal{S},j,u and vv, but is independent on ฮผ\mu, such that

(Gu~ฮผ(t),u~ฮผ(t)โˆ’v)Xโ‰คcฮผโˆ€vโˆˆX\left(G\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-v\right)_{X}\leq c\mu\quad\forall v\in X (4.9)

We take now v=u~(t)v=\widetilde{u}(t) in (4.9), then we pass to the upper limit as ฮผโ†’0\mu\rightarrow 0 in the resulting inequality to obtain

limฮผโ†’0sup(Gu~ฮผ(t),u~ฮผ(t)โˆ’u~(t))Xโ‰ค0\lim_{\mu\rightarrow 0}\sup\left(G\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-\widetilde{u}(t)\right)_{X}\leq 0

Therefore, using assumption (3.8) (a), (b) the convergence (4.7) and standard arguments on pseudomonotone operators (see Proposition 1.23 in [22, for instance) we deduce that

limฮผโ†’0inf(Gu~ฮผ(t),u~ฮผ(t)โˆ’v)Xโ‰ฅ(Gu~(t),u~(t)โˆ’v)Xโˆ€vโˆˆX\lim_{\mu\rightarrow 0}\inf\left(G\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-v\right)_{X}\geq(G\widetilde{u}(t),\widetilde{u}(t)-v)_{X}\quad\forall v\in X (4.10)

On the other hand, the inequality (4.9) implies that

lim infฮผโ†’0(Gu~ฮผ(t),u~ฮผ(t)โˆ’v)Xโ‰ค0โˆ€vโˆˆX\liminf_{\mu\rightarrow 0}\left(G\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-v\right)_{X}\leq 0\quad\forall v\in X (4.11)

We combine the inequalities (4.10) and (4.11) to see that

(Gu~(t),u~(t)โˆ’v)Xโ‰ค0โˆ€vโˆˆX(G\widetilde{u}(t),\widetilde{u}(t)-v)_{X}\leq 0\quad\forall v\in X

and, taking v=u~(t)โˆ’Gu~(t)v=\widetilde{u}(t)-G\widetilde{u}(t) in this inequality yields โ€–Gu~(t)โ€–X2โ‰ค0\|G\widetilde{u}(t)\|_{X}^{2}\leq 0. We conclude that Gu~(t)=0XG\widetilde{u}(t)=0_{X} and, using assumption (3.8)(d) it follows that

u~(t)โˆˆK.\widetilde{u}(t)\in K. (4.12)

Next, from inequality (4.1) and assumption (3.8) (c) we find that

(Au~ฮผ(t),vโˆ’u~ฮผ(t))X+(๐’ฎu(t),vโˆ’u~ฮผ(t))X\displaystyle\left(A\widetilde{u}_{\mu}(t),v-\widetilde{u}_{\mu}(t)\right)_{X}+\left(\mathcal{S}u(t),v-\widetilde{u}_{\mu}(t)\right)_{X} (4.13)
+j(v)โˆ’j(u~ฮผ(t))โ‰ฅ(f(t),vโˆ’u~ฮผ(t))Xโˆ€vโˆˆK\displaystyle\quad+j(v)-j\left(\widetilde{u}_{\mu}(t)\right)\geq\left(f(t),v-\widetilde{u}_{\mu}(t)\right)_{X}\quad\forall v\in K

We now take v=u~(t)โˆˆKv=\widetilde{u}(t)\in K in (4.13) and obtain

(Au~ฮผ(t),u~ฮผ(t)โˆ’u~(t))Xโ‰ค(๐’ฎu(t),u~(t)โˆ’u~ฮผ(t))X\displaystyle\left(A\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-\widetilde{u}(t)\right)_{X}\leq\left(\mathcal{S}u(t),\widetilde{u}(t)-\widetilde{u}_{\mu}(t)\right)_{X}
+j(u~(t))โˆ’j(u~ฮผ(t))+(f(t),u~ฮผ(t)โˆ’u~(t))X\displaystyle\quad+j(\widetilde{u}(t))-j\left(\widetilde{u}_{\mu}(t)\right)+\left(f(t),\widetilde{u}_{\mu}(t)-\widetilde{u}(t)\right)_{X}

then we pass to the upper limit as ฮผโ†’0\mu\rightarrow 0 in this inequality and use the weak convergence (4.7) and the assumption (3.5). As a result we obtain

limฮผโ†’0sup(Au~ฮผ(t),u~ฮผ(t)โˆ’u~(t))Xโ‰ค0\lim_{\mu\rightarrow 0}\sup\left(A\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-\widetilde{u}(t)\right)_{X}\leq 0 (4.14)

and, using again the argument on pseudomonotonicity employed in the proof of Lemma 4.3, it follows that

lim infฮผโ†’0(Au~ฮผ(t),u~ฮผ(t)โˆ’v)Xโ‰ฅ(Au~(t),u~(t)โˆ’v)Xโˆ€vโˆˆX\liminf_{\mu\rightarrow 0}\left(A\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-v\right)_{X}\geq(A\widetilde{u}(t),\widetilde{u}(t)-v)_{X}\quad\forall v\in X (4.15)

On the other hand, passing to the lower limit as ฮผโ†’0\mu\rightarrow 0 in (4.13) and using (4.7) yields

limฮผโ†’0inf(Au~ฮผ(t),u~ฮผ(t)โˆ’v)Xโ‰ค(๐’ฎu(t),vโˆ’u~(t))X\displaystyle\lim_{\mu\rightarrow 0}\inf\left(A\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-v\right)_{X}\leq(\mathcal{S}u(t),v-\widetilde{u}(t))_{X} (4.16)
+j(v)โˆ’j(u~(t))+(f(t),u~(t)โˆ’v)Xโˆ€vโˆˆK.\displaystyle\quad+j(v)-j(\widetilde{u}(t))+(f(t),\widetilde{u}(t)-v)_{X}\quad\forall v\in K.

We combine now the inequalities (4.15) and (4.16) to see that

(Au~(t),vโˆ’u~(t))X+(๐’ฎu(t),vโˆ’u~(t))X\displaystyle(A\widetilde{u}(t),v-\widetilde{u}(t))_{X}+(\mathcal{S}u(t),v-\widetilde{u}(t))_{X} (4.17)
+j(v)โˆ’j(u~(t))โ‰ฅ(f(t),vโˆ’u~(t))Xโˆ€vโˆˆK\displaystyle\quad+j(v)-j(\widetilde{u}(t))\geq(f(t),v-\widetilde{u}(t))_{X}\quad\forall v\in K

Next, we take v=u(t)v=u(t) in (4.17) and v=u~(t)v=\widetilde{u}(t) in (3.7). Then, adding the resulting inequalities and using the strong monotonicity of the operator AA we obtain that

u~(t)=u(t)\widetilde{u}(t)=u(t) (4.18)

which concludes the proof.
The next step is provided by the following weak convergence result.

Lemma 4.4 For each tโˆˆโ„+t\in\mathbb{R}_{+}the whole sequence {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\} converges weakly in XX to u(t)u(t) as ฮผโ†’0\mu\rightarrow 0.

Proof. Let tโˆˆโ„+t\in\mathbb{R}_{+}. A carefully examination of the proof of Lemma 4.3 shows that any weak convergent subsequence of the sequence {u~ฮผ(t)}โŠ‚X\left\{\widetilde{u}_{\mu}(t)\right\}\subset X converges weakly to u(t)u(t), where, recall, u(t)u(t) is the element of XX which solves the variational inequality (3.7) at the moment tt. This inequality has a unique solution and, moreover, estimate (4.6) shows that the sequence {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\} is bounded in XX. Lemma 4.4 is now a consequence of a standard compactness argument.

We proceed with the following strong convergence result.
Lemma 4.5 For each tโˆˆโ„+t\in\mathbb{R}_{+}the sequence {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\} converges strongly in XX to u(t)u(t), that is

u~ฮผ(t)โ†’u(t) in X as ฮผโ†’0.\widetilde{u}_{\mu}(t)\rightarrow u(t)\quad\text{ in }\quad X\quad\text{ as }\quad\mu\rightarrow 0. (4.19)

Proof. Let ฮผ>0\mu>0 and tโˆˆโ„+t\in\mathbb{R}_{+}. We take v=u~(t)v=\widetilde{u}(t) in (4.15) to see that

lim infฮผโ†’0(Au~ฮผ(t),u~ฮผ(t)โˆ’u~(t))Xโ‰ฅ0\liminf_{\mu\rightarrow 0}\left(A\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-\widetilde{u}(t)\right)_{X}\geq 0

then we combine this inequality with (4.14) to obtain that

limฮผโ†’0(Au~ฮผ(t),u~ฮผ(t)โˆ’u~(t))X=0\lim_{\mu\rightarrow 0}\left(A\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-\widetilde{u}(t)\right)_{X}=0

Finally, we use (4.18) to find that

limฮผโ†’0(Au~ฮผ(t),u~ฮผ(t)โˆ’u(t))X=0\lim_{\mu\rightarrow 0}\left(A\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-u(t)\right)_{X}=0 (4.20)

On the other hand, from the weak convergence of the sequence {u~ฮผ(t)}\left\{\widetilde{u}_{\mu}(t)\right\} to u(t)u(t), guaranteed by Lemma 4.4, it follows that

limฮผโ†’0(Au(t),u~ฮผ(t)โˆ’u(t))X=0\lim_{\mu\rightarrow 0}\left(Au(t),\widetilde{u}_{\mu}(t)-u(t)\right)_{X}=0 (4.21)

Next, from the strong monotonicity of the operator AA we have

mโ€–u~ฮผ(t)โˆ’u(t)โ€–2โ‰ค(Au~ฮผ(t)โˆ’Au(t),u~ฮผ(t)โˆ’u(t))X\displaystyle m\left\|\widetilde{u}_{\mu}(t)-u(t)\right\|^{2}\leq\left(A\widetilde{u}_{\mu}(t)-Au(t),\widetilde{u}_{\mu}(t)-u(t)\right)_{X} (4.22)
=(Au~ฮผ(t),u~ฮผ(t)โˆ’u(t))Xโˆ’(Au(t),u~ฮผ(t)โˆ’u(t))X\displaystyle\quad=\left(A\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-u(t)\right)_{X}-\left(Au(t),\widetilde{u}_{\mu}(t)-u(t)\right)_{X}

The strong convergence (4.19) is now a consequence of (4.20)-(4.22).
The last step is provided by the following strong convergence result.
Lemma 4.6 For each tโˆˆโ„+t\in\mathbb{R}_{+}the sequence {uฮผ(t)}\left\{u_{\mu}(t)\right\} converges strongly in XX to u(t)u(t), that is

uฮผ(t)โ†’u(t) in X as ฮผโ†’0u_{\mu}(t)\rightarrow u(t)\quad\text{ in }\quad X\quad\text{ as }\quad\mu\rightarrow 0 (4.23)

Proof. Let tโˆˆโ„+t\in\mathbb{R}_{+}and nโˆˆโ„•โˆ—n\in\mathbb{N}^{*} be such that tโˆˆ[0,n]t\in[0,n]. Let also ฮผ>0\mu>0. We take v=uฮผ(t)v=u_{\mu}(t) in (4.1) and v=u~ฮผ(t)v=\widetilde{u}_{\mu}(t) in (3.9). Then, adding the resulting inequalities we deduce that

(Auฮผ(t)\displaystyle\left(Au_{\mu}(t)\right. โˆ’Au~ฮผ(t),u~ฮผ(t)โˆ’uฮผ(t))X+(๐’ฎuฮผ(t)โˆ’๐’ฎu(t),u~ฮผ(t)โˆ’uฮผ(t))X\displaystyle\left.-A\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-u_{\mu}(t)\right)_{X}+\left(\mathcal{S}u_{\mu}(t)-\mathcal{S}u(t),\widetilde{u}_{\mu}(t)-u_{\mu}(t)\right)_{X}
+1ฮผ(Guฮผ(t)โˆ’Gu~ฮผ(t),u~ฮผ(t)โˆ’uฮผ(t))Xโ‰ฅ0\displaystyle+\frac{1}{\mu}\left(Gu_{\mu}(t)-G\widetilde{u}_{\mu}(t),\widetilde{u}_{\mu}(t)-u_{\mu}(t)\right)_{X}\geq 0

Next, we use the monotony of the operator GG, (3.8)(a), to obtain that

(Auฮผ(t)โˆ’Au~ฮผ(t),uฮผ(t)โˆ’u~ฮผ(t))Xโ‰ค(๐’ฎuฮผ(t)โˆ’๐’ฎu(t),u~ฮผ(t)โˆ’uฮผ(t))X\left(Au_{\mu}(t)-A\widetilde{u}_{\mu}(t),u_{\mu}(t)-\widetilde{u}_{\mu}(t)\right)_{X}\leq\left(\mathcal{S}u_{\mu}(t)-\mathcal{S}u(t),\widetilde{u}_{\mu}(t)-u_{\mu}(t)\right)_{X}

Therefore, using (3.3) (a) yields

โ€–uฮผ(t)โˆ’u~ฮผ(t)โ€–Xโ‰ค1mโ€–๐’ฎuฮผ(t)โˆ’๐’ฎu(t)โ€–X\left\|u_{\mu}(t)-\widetilde{u}_{\mu}(t)\right\|_{X}\leq\frac{1}{m}\left\|\mathcal{S}u_{\mu}(t)-\mathcal{S}u(t)\right\|_{X} (4.24)

We now combine (4.24) and (3.4) to find that

โ€–uฮผ(t)โˆ’u~ฮผ(t)โ€–Xโ‰คdnmโˆซ0tโ€–uฮผ(s)โˆ’u(s)โ€–X๐‘‘s\left\|u_{\mu}(t)-\widetilde{u}_{\mu}(t)\right\|_{X}\leq\frac{d_{n}}{m}\int_{0}^{t}\left\|u_{\mu}(s)-u(s)\right\|_{X}ds

It follows from here that

โ€–uฮผ(t)โˆ’u(t)โ€–Xโ‰คโ€–u~ฮผ(t)โˆ’u(t)โ€–X+dnmโˆซ0tโ€–uฮผ(s)โˆ’u(s)โ€–X๐‘‘s\left\|u_{\mu}(t)-u(t)\right\|_{X}\leq\left\|\widetilde{u}_{\mu}(t)-u(t)\right\|_{X}+\frac{d_{n}}{m}\int_{0}^{t}\left\|u_{\mu}(s)-u(s)\right\|_{X}ds

and, using a Gronwallโ€™s argument, we obtain that

โ€–uฮผ(t)โˆ’u(t)โ€–Xโ‰คโ€–u~ฮผ(t)โˆ’u(t)โ€–X+dnmโˆซ0tednm(tโˆ’s)โ€–u~ฮผ(s)โˆ’u(s)โ€–X๐‘‘s\left\|u_{\mu}(t)-u(t)\right\|_{X}\leq\left\|\widetilde{u}_{\mu}(t)-u(t)\right\|_{X}+\frac{d_{n}}{m}\int_{0}^{t}e^{\frac{d_{n}}{m}(t-s)}\left\|\widetilde{u}_{\mu}(s)-u(s)\right\|_{X}ds (4.25)

Note that ednm(tโˆ’s)โ‰คednmtโ‰คendnme^{\frac{d_{n}}{m}(t-s)}\leq e^{\frac{d_{n}}{m}t}\leq e^{\frac{nd_{n}}{m}} for all sโˆˆ[0,n]s\in[0,n] and, therefore, (4.25) yields

โ€–uฮผ(t)โˆ’u(t)โ€–Xโ‰คโ€–u~ฮผ(t)โˆ’u(t)โ€–X+dnmendnmโˆซ0tโ€–u~ฮผ(s)โˆ’u(s)โ€–X๐‘‘s\left\|u_{\mu}(t)-u(t)\right\|_{X}\leq\left\|\widetilde{u}_{\mu}(t)-u(t)\right\|_{X}+\frac{d_{n}}{m}e^{\frac{nd_{n}}{m}}\int_{0}^{t}\left\|\widetilde{u}_{\mu}(s)-u(s)\right\|_{X}ds (4.26)

On the other hand, by estimate (4.6), Lemma 4.5 and Lebesgueโ€™s convergence theorem it follows that

โˆซ0tโ€–u~ฮผ(s)โˆ’u(s)โ€–X๐‘‘sโ†’0 as ฮผโ†’0\int_{0}^{t}\left\|\widetilde{u}_{\mu}(s)-u(s)\right\|_{X}ds\rightarrow 0\quad\text{ as }\quad\mu\rightarrow 0 (4.27)

We use now (4.26), (4.27) and (4.19) to obtain the convergence (4.23), which concludes the proof.

We end this section with the remark that the points 1) and 2) of Theorem 3.2 correspond to Lemmas 4.1 and 4.6, respectively. Therefore, we conclude from here that the proof of Theorem 3.2 is complete.

5 Existence and uniqueness

We turn now to the variational analysis of problem ๐’ฌ\mathcal{Q}. To this end, we need further notation and preliminaries. First, we use the notation ๐’™=(xi)\boldsymbol{x}=\left(x_{i}\right) for a typical point in ฮฉโˆชฮ“\Omega\cup\Gamma and we denote by ๐‚=(ฮฝi)\boldsymbol{\nu}=\left(\nu_{i}\right) the outward unit normal at ฮ“\Gamma. Here and below the indices i,j,k,li,j,k,l run between 1 and dd and, unless stated otherwise, the summation convention over repeated indices is used. An index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. ui,j=โˆ‚ui/โˆ‚xju_{i,j}=\partial u_{i}/\partial x_{j}. Recall that the inner product and norm on โ„d\mathbb{R}^{d} and ๐•Šd\mathbb{S}^{d} are defined by

๐’–โ‹…๐’—=uivi,โ€–๐’—โ€–=(๐’—โ‹…๐’—)12โˆ€๐’–,๐’—โˆˆโ„d๐ˆโ‹…๐‰=ฯƒijฯ„ij,โ€–๐‰โ€–=(๐‰โ‹…๐‰)12โˆ€๐ˆ,๐‰โˆˆ๐•Šd\begin{array}[]{lrrl}\boldsymbol{u}\cdot\boldsymbol{v}=u_{i}v_{i},&\|\boldsymbol{v}\|=(\boldsymbol{v}\cdot\boldsymbol{v})^{\frac{1}{2}}&\forall\boldsymbol{u},\boldsymbol{v}\in\mathbb{R}^{d}\\ \boldsymbol{\sigma}\cdot\boldsymbol{\tau}=\sigma_{ij}\tau_{ij},&\|\boldsymbol{\tau}\|=(\boldsymbol{\tau}\cdot\boldsymbol{\tau})^{\frac{1}{2}}&\forall\boldsymbol{\sigma},\boldsymbol{\tau}\in\mathbb{S}^{d}\end{array}

We use standard notation for the Lebesgue and Sobolev spaces associated to ฮฉ\Omega and ฮ“\Gamma and, moreover, we consider the following spaces:

V={๐’—=(vi)โˆˆH1(ฮฉ)d:vi=0 on ฮ“1}\displaystyle V=\left\{\boldsymbol{v}=\left(v_{i}\right)\in H^{1}(\Omega)^{d}:v_{i}=0\text{ on }\Gamma_{1}\right\}
Q={๐‰=(ฯ„ij):ฯ„ij=ฯ„jiโˆˆL2(ฮฉ)}\displaystyle Q=\left\{\boldsymbol{\tau}=\left(\tau_{ij}\right):\tau_{ij}=\tau_{ji}\in L^{2}(\Omega)\right\}
Q1={๐‰โˆˆQ:ฯ„ij,jโˆˆL2(ฮฉ)}\displaystyle Q_{1}=\left\{\boldsymbol{\tau}\in Q:\tau_{ij,j}\in L^{2}(\Omega)\right\}

These are real Hilbert spaces endowed with the inner products

(๐’–,๐’—)V=โˆซฮฉ๐œบ(๐’–)โ‹…๐œบ(๐’—)๐‘‘x,(๐ˆ,๐‰)Q=โˆซฮฉ๐ˆโ‹…๐‰๐‘‘x\displaystyle(\boldsymbol{u},\boldsymbol{v})_{V}=\int_{\Omega}\boldsymbol{\varepsilon}(\boldsymbol{u})\cdot\boldsymbol{\varepsilon}(\boldsymbol{v})dx,\quad(\boldsymbol{\sigma},\boldsymbol{\tau})_{Q}=\int_{\Omega}\boldsymbol{\sigma}\cdot\boldsymbol{\tau}dx
(๐ˆ,๐‰)Q1=(๐ˆ,๐‰)Q+(Divโก๐ˆ,Divโก๐‰)L2(ฮฉ)d\displaystyle(\boldsymbol{\sigma},\boldsymbol{\tau})_{Q_{1}}=(\boldsymbol{\sigma},\boldsymbol{\tau})_{Q}+(\operatorname{Div}\boldsymbol{\sigma},\operatorname{Div}\boldsymbol{\tau})_{L^{2}(\Omega)^{d}}

Here and below ๐œบ\boldsymbol{\varepsilon} and Div are the deformation and the divergence operators, respectively, defined by

๐œบ(๐’–)=(ฮตij(๐’–)),ฮตij(๐’–)=12(ui,j+uj,i),Divโก๐ˆ=(ฯƒij,j).\boldsymbol{\varepsilon}(\boldsymbol{u})=\left(\varepsilon_{ij}(\boldsymbol{u})\right),\quad\varepsilon_{ij}(\boldsymbol{u})=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right),\quad\operatorname{Div}\boldsymbol{\sigma}=\left(\sigma_{ij,j}\right).

Completeness of the space ( V,โˆฅโ‹…โˆฅVV,\|\cdot\|_{V} ) follows from the assumption meas (ฮ“1)>0\left(\Gamma_{1}\right)>0, which allows the use of Kornโ€™s inequality.

For an element ๐’—โˆˆV\boldsymbol{v}\in V we still write ๐’—\boldsymbol{v} for the trace of ๐’—\boldsymbol{v} on the boundary and we denote by vฮฝv_{\nu} and ๐’—ฯ„\boldsymbol{v}_{\tau} the normal and tangential components of ๐’—\boldsymbol{v} on ฮ“\Gamma, given by vฮฝ=๐’—โ‹…๐‚,๐’—ฯ„=๐’—โˆ’vฮฝ๐‚v_{\nu}=\boldsymbol{v}\cdot\boldsymbol{\nu},\boldsymbol{v}_{\tau}=\boldsymbol{v}-v_{\nu}\boldsymbol{\nu}. Let ฮ“3\Gamma_{3} be a measurable part of ฮ“\Gamma. Then, by the Sobolev trace theorem, there exists a positive constant c0c_{0} which depends on ฮฉ,ฮ“1\Omega,\Gamma_{1} and ฮ“3\Gamma_{3} such that

โ€–๐’—โ€–L2(ฮ“3)dโ‰คc0โ€–๐’—โ€–Vโˆ€๐’—โˆˆV.\|\boldsymbol{v}\|_{L^{2}\left(\Gamma_{3}\right)^{d}}\leq c_{0}\|\boldsymbol{v}\|_{V}\quad\forall\boldsymbol{v}\in V. (5.1)

Also, for a regular function ๐ˆโˆˆQ\boldsymbol{\sigma}\in Q we use the notation ฯƒฮฝ\sigma_{\nu} and ๐ˆฯ„\boldsymbol{\sigma}_{\tau} for the normal and the tangential trace, i.e. ฯƒฮฝ=(๐ˆ๐‚)โ‹…๐‚\sigma_{\nu}=(\boldsymbol{\sigma}\boldsymbol{\nu})\cdot\boldsymbol{\nu} and ๐ˆฯ„=๐ˆ๐‚โˆ’ฯƒฮฝ๐‚\boldsymbol{\sigma}_{\tau}=\boldsymbol{\sigma}\boldsymbol{\nu}-\sigma_{\nu}\boldsymbol{\nu}. Moreover, we recall that
the following Greenโ€™s formula holds:

โˆซฮฉ๐ˆโ‹…๐œบ(๐’—)๐‘‘x+โˆซฮฉDivโก๐ˆโ‹…๐’—dx=โˆซฮ“๐ˆ๐‚โ‹…๐’—๐‘‘aโˆ€๐’—โˆˆV\int_{\Omega}\boldsymbol{\sigma}\cdot\boldsymbol{\varepsilon}(\boldsymbol{v})dx+\int_{\Omega}\operatorname{Div}\boldsymbol{\sigma}\cdot\boldsymbol{v}dx=\int_{\Gamma}\boldsymbol{\sigma}\boldsymbol{\nu}\cdot\boldsymbol{v}da\quad\forall\boldsymbol{v}\in V (5.2)

Finally, we denote by ๐โˆž\mathbf{Q}_{\infty} the space of fourth order tensor fields given by

๐โˆž={โ„ฐ=(โ„ฐijkl):โ„ฐijkl=โ„ฐjikl=โ„ฐklijโˆˆLโˆž(ฮฉ),1โ‰คi,j,k,lโ‰คd},\mathbf{Q}_{\infty}=\left\{\mathcal{E}=\left(\mathcal{E}_{ijkl}\right):\mathcal{E}_{ijkl}=\mathcal{E}_{jikl}=\mathcal{E}_{klij}\in L^{\infty}(\Omega),\quad 1\leq i,j,k,l\leq d\right\},

and we recall that ๐โˆž\mathbf{Q}_{\infty} is a real Banach space with the norm

โ€–โ„ฐโ€–๐โˆž=max1โ‰คi,j,k,lโ‰คdโกโ€–โ„ฐijklโ€–Lโˆž(ฮฉ).\|\mathcal{E}\|_{\mathbf{Q}_{\infty}}=\max_{1\leq i,j,k,l\leq d}\left\|\mathcal{E}_{ijkl}\right\|_{L^{\infty}(\Omega)}.

Moreover, a simple calculation shows that

โ€–โ„ฐ๐‰โ€–Qโ‰คโ€–โ„ฐโ€–๐โˆžโ€–๐‰โ€–Qโˆ€โ„ฐโˆˆ๐โˆž,๐‰โˆˆQ.\|\mathcal{E}\boldsymbol{\tau}\|_{Q}\leq\|\mathcal{E}\|_{\mathbf{Q}_{\infty}}\|\boldsymbol{\tau}\|_{Q}\quad\forall\mathcal{E}\in\mathbf{Q}_{\infty},\boldsymbol{\tau}\in Q. (5.3)

Next, we list the assumptions on the data, derive the variational formulation of the problem ๐’ฌ\mathcal{Q} and then we state and prove its unique weak solvability. To this end we assume that the elasticity operator ๐’œ\mathcal{A} and the relaxation tensor โ„ฌ\mathcal{B} satisfy the following conditions.

{ (a) ๐’œ:ฮฉร—๐•Šdโ†’๐•Šd (b) There exists L๐’œ>0 such that โ€–๐’œ(๐’™,๐œบ1)โˆ’๐’œ(๐’™,๐œบ2)โ€–โ‰คL๐’œโ€–๐œบ1โˆ’๐œบ2โ€–โˆ€๐œบ1,๐œบ2โˆˆ๐•Šd, a.e. ๐’™โˆˆฮฉ (c) There exists m๐’œ>0 such that (๐’œ(๐’™,๐œบ1)โˆ’๐’œ(๐’™,๐œบ2))โ‹…(๐œบ1โˆ’๐œบ2)โ‰ฅm๐’œโ€–๐œบ1โˆ’๐œบ2โ€–2โˆ€๐œบ1,๐œบ2โˆˆ๐•Šd, a.e. ๐’™โˆˆฮฉ. (d) The mapping ๐’™โ†ฆ๐’œ(๐’™,๐œบ) is measurable on ฮฉ, for any ๐œบโˆˆ๐•Šd (e) The mapping ๐’™โ†ฆ๐’œ(๐’™,๐ŸŽ) belongs to Q.โ„ฌโˆˆC(โ„+;๐โˆž).\left\{\begin{array}[]{l}\text{ (a) }\mathcal{A}:\Omega\times\mathbb{S}^{d}\rightarrow\mathbb{S}^{d}\text{. }\\ \text{ (b) There exists }L_{\mathcal{A}}>0\text{ such that }\\ \left\|\mathcal{A}\left(\boldsymbol{x},\boldsymbol{\varepsilon}_{1}\right)-\mathcal{A}\left(\boldsymbol{x},\boldsymbol{\varepsilon}_{2}\right)\right\|\leq L_{\mathcal{A}}\left\|\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right\|\\ \quad\forall\boldsymbol{\varepsilon}_{1},\boldsymbol{\varepsilon}_{2}\in\mathbb{S}^{d}\text{, a.e. }\boldsymbol{x}\in\Omega\text{. }\\ \text{ (c) There exists }m_{\mathcal{A}}>0\text{ such that }\\ \quad\left(\mathcal{A}\left(\boldsymbol{x},\boldsymbol{\varepsilon}_{1}\right)-\mathcal{A}\left(\boldsymbol{x},\boldsymbol{\varepsilon}_{2}\right)\right)\cdot\left(\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right)\geq m_{\mathcal{A}}\left\|\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right\|^{2}\\ \quad\forall\boldsymbol{\varepsilon}_{1},\boldsymbol{\varepsilon}_{2}\in\mathbb{S}^{d},\text{ a.e. }\boldsymbol{x}\in\Omega.\\ \text{ (d) The mapping }\boldsymbol{x}\mapsto\mathcal{A}(\boldsymbol{x},\boldsymbol{\varepsilon})\text{ is measurable on }\Omega,\\ \text{ for any }\boldsymbol{\varepsilon}\in\mathbb{S}^{d}\text{. }\\ \text{ (e) The mapping }\boldsymbol{x}\mapsto\mathcal{A}(\boldsymbol{x},\mathbf{0})\text{ belongs to }Q.\\ \qquad\mathcal{B}\in C\left(\mathbb{R}_{+};\mathbf{Q}_{\infty}\right).\end{array}\right.

The densities of body forces and surface tractions are such that

๐’‡0โˆˆC(โ„+;L2(ฮฉ)d),๐’‡2โˆˆC(โ„+;L2(ฮ“2)d).\boldsymbol{f}_{0}\in C\left(\mathbb{R}_{+};L^{2}(\Omega)^{d}\right),\quad\boldsymbol{f}_{2}\in C\left(\mathbb{R}_{+};L^{2}\left(\Gamma_{2}\right)^{d}\right). (5.6)

Finally, the normal compliance function pp and the surface yield function FF satisfy

{ (a) p:โ„โ†’โ„+.(b) There exists Lp>0 such that |p(r1)โˆ’p(r2)|โ‰คLp|r1โˆ’r2|โˆ€r1,r2โˆˆโ„. (c) (p(r1)โˆ’p(r2))(r1โˆ’r2)โ‰ฅ0โˆ€r1,r2โˆˆโ„. (d) p(r)=0 iff rโ‰ค0.\left\{\begin{array}[]{l}\text{ (a) }p:\mathbb{R}\rightarrow\mathbb{R}_{+}.\\ \text{(b) There exists }L_{p}>0\text{ such that }\\ \quad\left|p\left(r_{1}\right)-p\left(r_{2}\right)\right|\leq L_{p}\left|r_{1}-r_{2}\right|\quad\forall r_{1},r_{2}\in\mathbb{R}.\\ \text{ (c) }\left(p\left(r_{1}\right)-p\left(r_{2}\right)\right)\left(r_{1}-r_{2}\right)\geq 0\quad\forall r_{1},r_{2}\in\mathbb{R}.\\ \text{ (d) }p(r)=0\text{ iff }\quad r\leq 0.\end{array}\right.
FโˆˆL2(ฮ“3),F(x)โ‰ฅ0 a.e. xโˆˆฮ“3.F\in L^{2}\left(\Gamma_{3}\right),\quad F(x)\geq 0\text{ a.e. }x\in\Gamma_{3}. (5.8)

In what follows we consider the set of admissible displacements defined by

U={๐’—โˆˆV:vฮฝโ‰คg on ฮ“3}.U=\left\{\boldsymbol{v}\in V:v_{\nu}\leq g\text{ on }\Gamma_{3}\right\}. (5.9)

Moreover, we define the operator P:Vโ†’VP:V\rightarrow V and the functions j:Vโ†’โ„+j:V\rightarrow\mathbb{R}_{+}, ๐’‡:โ„+โ†’V\boldsymbol{f}:\mathbb{R}_{+}\rightarrow V by equalities

(P๐’–,๐’—)V=โˆซฮ“3p(uฮฝ)vฮฝ๐‘‘aโˆ€๐’–,๐’—โˆˆV\displaystyle(P\boldsymbol{u},\boldsymbol{v})_{V}=\int_{\Gamma_{3}}p\left(u_{\nu}\right)v_{\nu}da\quad\forall\boldsymbol{u},\boldsymbol{v}\in V (5.10)
j(๐’—)=โˆซฮ“3Fvฮฝ+๐‘‘aโˆ€๐’—โˆˆV\displaystyle j(\boldsymbol{v})=\int_{\Gamma_{3}}Fv_{\nu}^{+}da\quad\forall\boldsymbol{v}\in V (5.11)
(๐’‡(t),๐’—)V=โˆซฮฉ๐’‡0(t)โ‹…๐’—๐‘‘x+โˆซฮ“2๐’‡2(t)โ‹…๐’—๐‘‘aโˆ€๐’—โˆˆV,tโˆˆ[0,T]\displaystyle(\boldsymbol{f}(t),\boldsymbol{v})_{V}=\int_{\Omega}\boldsymbol{f}_{0}(t)\cdot\boldsymbol{v}dx+\int_{\Gamma_{2}}\boldsymbol{f}_{2}(t)\cdot\boldsymbol{v}da\quad\forall\boldsymbol{v}\in V,t\in[0,T] (5.12)

Here and below, for rโˆˆโ„r\in\mathbb{R} we denote by r+r^{+}its positive part, i.e. r+=maxโก{r,0}r^{+}=\max\{r,0\}. Note that assumptions (5.6)-(5.8) imply that the integrals in (5.10)-(5.12) are welldefined.

Assume in what follows that ( ๐’–,๐ˆ\boldsymbol{u},\boldsymbol{\sigma} ) are sufficiently regular functions which satisfy (2.1)-(2.6) and let ๐’—โˆˆU\boldsymbol{v}\in U and t>0t>0 be given. First, we use Greenโ€™s formula (5.2) and the equilibrium equation (2.2) to see that

โˆซฮฉ๐ˆ(t)โ‹…(๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))๐‘‘x=โˆซฮฉ๐’‡0(t)โ‹…(๐’—โˆ’๐’–(t))๐‘‘x+โˆซฮ“๐ˆ(t)๐‚โ‹…(๐’—โˆ’๐’–(t))๐‘‘a\int_{\Omega}\boldsymbol{\sigma}(t)\cdot(\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))dx=\int_{\Omega}\boldsymbol{f}_{0}(t)\cdot(\boldsymbol{v}-\boldsymbol{u}(t))dx+\int_{\Gamma}\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\boldsymbol{v}-\boldsymbol{u}(t))da

We split the surface integral over ฮ“1,ฮ“2\Gamma_{1},\Gamma_{2} and ฮ“3\Gamma_{3} and, since ๐’—โˆ’๐’–(t)=๐ŸŽ\boldsymbol{v}-\boldsymbol{u}(t)=\mathbf{0} a.e. on ฮ“1\Gamma_{1}, ๐ˆ(t)๐‚=๐’‡2(t)\boldsymbol{\sigma}(t)\boldsymbol{\nu}=\boldsymbol{f}_{2}(t) on ฮ“2\Gamma_{2}, we deduce that

โˆซฮฉ๐ˆ(t)โ‹…(๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))๐‘‘x=โˆซฮฉ๐’‡0(t)โ‹…(๐’—โˆ’๐’–(t))๐‘‘x\displaystyle\int_{\Omega}\boldsymbol{\sigma}(t)\cdot(\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))dx=\int_{\Omega}\boldsymbol{f}_{0}(t)\cdot(\boldsymbol{v}-\boldsymbol{u}(t))dx
+โˆซฮ“2๐’‡2(t)โ‹…(๐’—โˆ’๐’–(t))๐‘‘a+โˆซฮ“3๐ˆ(t)๐‚โ‹…(๐’—โˆ’๐’–(t))๐‘‘a\displaystyle\quad+\int_{\Gamma_{2}}\boldsymbol{f}_{2}(t)\cdot(\boldsymbol{v}-\boldsymbol{u}(t))da+\int_{\Gamma_{3}}\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\boldsymbol{v}-\boldsymbol{u}(t))da

Moreover, since

๐ˆ(t)๐‚โ‹…(๐’—โˆ’๐’–(t))=ฯƒฮฝ(t)(vฮฝโˆ’uฮฝ(t))+๐ˆฯ„(t)โ‹…(๐’—ฯ„โˆ’๐’–ฯ„(t)) on ฮ“3,\boldsymbol{\sigma}(t)\boldsymbol{\nu}\cdot(\boldsymbol{v}-\boldsymbol{u}(t))=\sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right)+\boldsymbol{\sigma}_{\tau}(t)\cdot\left(\boldsymbol{v}_{\tau}-\boldsymbol{u}_{\tau}(t)\right)\quad\text{ on }\Gamma_{3},

taking into account the frictionless condition (2.5) we obtain

โˆซฮฉ๐ˆ(t)โ‹…(๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))๐‘‘x=โˆซฮฉ๐’‡0(t)โ‹…(๐’—โˆ’๐’–(t))๐‘‘x\displaystyle\int_{\Omega}\boldsymbol{\sigma}(t)\cdot(\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))dx=\int_{\Omega}\boldsymbol{f}_{0}(t)\cdot(\boldsymbol{v}-\boldsymbol{u}(t))dx (5.13)
+โˆซฮ“2๐’‡2(t)โ‹…(๐’—โˆ’๐’–(t))๐‘‘a+โˆซฮ“3ฯƒฮฝ(t)(vฮฝโˆ’uฮฝ(t))๐‘‘a\displaystyle\quad+\int_{\Gamma_{2}}\boldsymbol{f}_{2}(t)\cdot(\boldsymbol{v}-\boldsymbol{u}(t))da+\int_{\Gamma_{3}}\sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right)da

We write now

ฯƒฮฝ(\displaystyle\sigma_{\nu}( (t)(vฮฝโˆ’uฮฝ(t))=(ฯƒฮฝ(t)+p(uฮฝ(t))+ฮพ(t))(vฮฝโˆ’g)\displaystyle(t)\left(v_{\nu}-u_{\nu}(t)\right)=\left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t)\right)\left(v_{\nu}-g\right)
+(ฯƒฮฝ(t)+p(uฮฝ(t))+ฮพ(t))(gโˆ’uฮฝ(t))\displaystyle+\left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t)\right)\left(g-u_{\nu}(t)\right)
โˆ’(p(uฮฝ(t))+ฮพ(t))(vฮฝโˆ’uฮฝ(t)) on ฮ“3\displaystyle-\left(p\left(u_{\nu}(t)\right)+\xi(t)\right)\left(v_{\nu}-u_{\nu}(t)\right)\quad\text{ on }\Gamma_{3}

then we use the contact conditions (2.6) and the definition (5.9) of the set UU to see that

ฯƒฮฝ(t)(vฮฝโˆ’uฮฝ(t))โ‰ฅโˆ’(p(uฮฝ(t))+ฮพ(t))(vฮฝโˆ’uฮฝ(t)) on ฮ“3.\sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right)\geq-\left(p\left(u_{\nu}(t)\right)+\xi(t)\right)\left(v_{\nu}-u_{\nu}(t)\right)\quad\text{ on }\Gamma_{3}. (5.14)

We use (2.6), again, and the hypothesis (5.8) on function FF to deduce that

F(vฮฝ+โˆ’uฮฝ+(t))โ‰ฅฮพ(t)(vฮฝโˆ’uฮฝ(t)) on ฮ“3.F\left(v_{\nu}^{+}-u_{\nu}^{+}(t)\right)\geq\xi(t)\left(v_{\nu}-u_{\nu}(t)\right)\quad\text{ on }\Gamma_{3}. (5.15)

Then we add the inequalities (5.14) and (5.15) and integrate the result on ฮ“3\Gamma_{3} to find that

โˆซฮ“3ฯƒฮฝ(t)(vฮฝโˆ’uฮฝ(t))๐‘‘a\displaystyle\int_{\Gamma_{3}}\sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right)da (5.16)
โ‰ฅโˆ’โˆซฮ“3p(uฮฝ(t))(vฮฝโˆ’uฮฝ(t))๐‘‘aโˆ’โˆซฮ“3F(vฮฝ+โˆ’uฮฝ+(t))๐‘‘a\displaystyle\quad\geq-\int_{\Gamma_{3}}p\left(u_{\nu}(t)\right)\left(v_{\nu}-u_{\nu}(t)\right)da-\int_{\Gamma_{3}}F\left(v_{\nu}^{+}-u_{\nu}^{+}(t)\right)da

Finally, we combine (5.13) and (5.16) and use the definitions (5.10)-(5.12) to deduce that

(๐ˆ(t),๐œบ(๐’—)โˆ’\displaystyle(\boldsymbol{\sigma}(t),\boldsymbol{\varepsilon}(\boldsymbol{v})- ๐œบ(๐’–(t)))Q+(P๐’–(t),๐’—โˆ’๐’–(t))V+j(๐’—)โˆ’j(๐’–(t))\displaystyle\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q}+(P\boldsymbol{u}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}+j(\boldsymbol{v})-j(\boldsymbol{u}(t)) (5.17)
โ‰ฅ(๐’‡(t),๐’—โˆ’๐’–(t))Vโˆ€๐’—โˆˆU.\displaystyle\geq(\boldsymbol{f}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}\quad\forall\boldsymbol{v}\in U.

We now substitute the constitutive law (2.1) in (5.17) to obtain the following variational formulation of Problem ๐’ฌ\mathcal{Q}.

Problem ๐’ฌV\mathcal{Q}^{V}. Find a displacement field ๐’–:โ„+โ†’U\boldsymbol{u}:\mathbb{R}_{+}\rightarrow U such that, for all tโˆˆโ„+t\in\mathbb{R}_{+}, the inequality below holds:

(๐’œ๐œบ(๐’–(t)),๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))Q+(โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–(s))๐‘‘s,๐œบ(๐’—)โˆ’๐œบ(๐’–(t)))Q\displaystyle(\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}(t)),\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q}+\left(\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}(\boldsymbol{u}(s))ds,\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t))\right)_{Q} (5.18)
+(P๐’–(t),๐’—โˆ’๐’–(t))V+j(๐’—)โˆ’j(๐’–(t))โ‰ฅ(๐’‡(t),๐’—โˆ’๐’–(t))Vโˆ€๐’—โˆˆU\displaystyle\quad+(P\boldsymbol{u}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}+j(\boldsymbol{v})-j(\boldsymbol{u}(t))\geq(\boldsymbol{f}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}\quad\forall\boldsymbol{v}\in U

In the study of the problem ๐’ฌV\mathcal{Q}^{V} we have the following existence and uniqueness result.

Theorem 5.1 Assume that (5.4)-(5.8) hold. Then, Problem ๐’ฌV\mathcal{Q}^{V} has a unique solution which satisfies ๐’–โˆˆC(โ„+;U)\boldsymbol{u}\in C\left(\mathbb{R}_{+};U\right).

Proof. To solve the variational inequality (5.18) we use Theorem 3.1 with X=VX=V and K=UK=U. To this end we consider the operators A:Vโ†’VA:V\rightarrow V and ๐’ฎ:C(โ„+;V)โ†’C(โ„+;V)\mathcal{S}:C\left(\mathbb{R}_{+};V\right)\rightarrow C\left(\mathbb{R}_{+};V\right) defined by

(A๐’–,๐’—)V=(๐’œ๐œบ(๐’–),๐œบ(๐’—))Q+(P๐’–,๐’—)Vโˆ€๐’–,๐’—โˆˆV\displaystyle(A\boldsymbol{u},\boldsymbol{v})_{V}=(\mathcal{A}\boldsymbol{\varepsilon}(\boldsymbol{u}),\boldsymbol{\varepsilon}(\boldsymbol{v}))_{Q}+(P\boldsymbol{u},\boldsymbol{v})_{V}\quad\forall\boldsymbol{u},\boldsymbol{v}\in V (5.19)
(๐’ฎ๐’–(t),๐’—)V=(โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–(s))๐‘‘s,๐œบ(๐’—))Q\displaystyle(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v})_{V}=\left(\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}(\boldsymbol{u}(s))ds,\boldsymbol{\varepsilon}(\boldsymbol{v})\right)_{Q} (5.20)
โˆ€๐’–โˆˆC(โ„+;V),๐’—โˆˆV\displaystyle\forall\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right),\boldsymbol{v}\in V

It is easy to see that condition (3.2) holds. Next, we use (5.4), (5.7) and (5.1) to see that the operator AA satisfies conditions (3.3) with M=L๐’œ+c02LpM=L_{\mathcal{A}}+c_{0}^{2}L_{p} and m=m๐’œm=m_{\mathcal{A}}. Let nโˆˆโ„•โˆ—n\in\mathbb{N}^{*}. Then, a simple calculation based on assumption (5.5) and inequality (5.3) shows that

โ€–๐’ฎ๐’–1(t)โˆ’๐’ฎ๐’–2(t)โ€–Vโ‰คmaxrโˆˆ[0,n]โกโ€–โ„ฌ(r)โ€–๐โˆžโˆซ0tโ€–๐’–1(s)โˆ’๐’–2(s)โ€–V๐‘‘s\displaystyle\left\|\mathcal{S}\boldsymbol{u}_{1}(t)-\mathcal{S}\boldsymbol{u}_{2}(t)\right\|_{V}\leq\max_{r\in[0,n]}\|\mathcal{B}(r)\|_{\mathbf{Q}_{\infty}}\int_{0}^{t}\left\|\boldsymbol{u}_{1}(s)-\boldsymbol{u}_{2}(s)\right\|_{V}ds (5.21)
โˆ€๐’–1,๐’–2โˆˆC(โ„+;V),โˆ€tโˆˆ[0,n]\displaystyle\forall\boldsymbol{u}_{1},\boldsymbol{u}_{2}\in C\left(\mathbb{R}_{+};V\right),\forall t\in[0,n]

This inequality shows that the operator ๐’ฎ\mathcal{S}, defined by (5.20), satisfies condition (3.4) with

dn=maxrโˆˆ[0,n]โกโ€–โ„ฌ(r)โ€–๐โˆžd_{n}=\max_{r\in[0,n]}\|\mathcal{B}(r)\|_{\mathbf{Q}_{\infty}}

Next, we use condition (5.8) to see that the functional jj defined by (5.11) is a seminorm on VV and, moreover, it satisfies

j(๐’—)โ‰คc0โ€–Fโ€–L2(ฮ“3)โ€–๐’—โ€–Vโˆ€๐’—โˆˆVj(\boldsymbol{v})\leq c_{0}\|F\|_{L^{2}\left(\Gamma_{3}\right)}\|\boldsymbol{v}\|_{V}\quad\forall\boldsymbol{v}\in V (5.22)

Inequality (5.22) shows that the seminorm jj is continuous on VV and, therefore, (3.5) holds. Finally, using assumption (5.6) and definition (5.12) we deduce that fโˆˆC(โ„+;V)f\in C\left(\mathbb{R}_{+};V\right) which shows that (3.6) holds, too.

It follows now from Theorem 3.1 that there exists a unique function ๐’–โˆˆC(โ„+;V)\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right) which satisfies the inequality

๐’–(t)โˆˆ\displaystyle\boldsymbol{u}(t)\in U,(A๐’–(t),๐’—โˆ’๐’–(t))V+(๐’ฎ๐’–(t),๐’—โˆ’๐’–(t))V\displaystyle U,\quad(A\boldsymbol{u}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}+(\mathcal{S}\boldsymbol{u}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V} (5.23)
+j(๐’—)โˆ’j(๐’–(t))โ‰ฅ(๐’‡(t),๐’—โˆ’๐’–(t))Vโˆ€๐’—โˆˆU\displaystyle+j(\boldsymbol{v})-j(\boldsymbol{u}(t))\geq(\boldsymbol{f}(t),\boldsymbol{v}-\boldsymbol{u}(t))_{V}\quad\forall\boldsymbol{v}\in U

for all tโˆˆโ„+t\in\mathbb{R}_{+}. And, using (5.19) and (5.20) we deduce that there exists a unique function ๐’–โˆˆC(โ„+;V)\boldsymbol{u}\in C\left(\mathbb{R}_{+};V\right) such that (5.18) holds for all tโˆˆโ„+t\in\mathbb{R}_{+}, which concludes the proof.

Let ๐ˆ\boldsymbol{\sigma} be the function defined by (2.1). Then, it follows from (5.4) and (5.5) that ๐ˆโˆˆC(โ„+;Q)\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q\right). Moreover, it is easy to see that (5.17) holds for all tโˆˆโ„+t\in\mathbb{R}_{+}and, using standard arguments, it results from here that

Divโก๐ˆ(t)+๐’‡0(t)=๐ŸŽโˆ€tโˆˆโ„+\operatorname{Div}\boldsymbol{\sigma}(t)+\boldsymbol{f}_{0}(t)=\mathbf{0}\quad\forall t\in\mathbb{R}_{+} (5.24)

Therefore, using the regularity ๐’‡0โˆˆC(โ„+;L2(ฮฉ)d)\boldsymbol{f}_{0}\in C\left(\mathbb{R}_{+};L^{2}(\Omega)^{d}\right) in (5.6) we deduce that Divโก๐ˆโˆˆC(โ„+;L2(ฮฉ)d)\operatorname{Div}\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};L^{2}(\Omega)^{d}\right) which implies that ๐ˆโˆˆC(โ„+;Q1)\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q_{1}\right). A couple of functions ( ๐’–,๐ˆ\boldsymbol{u},\boldsymbol{\sigma} ) which satisfies (2.1), (5.18) for all tโˆˆโ„+t\in\mathbb{R}_{+}is called a weak solution to the contact problem ๐’ฌ\mathcal{Q}. We conclude that Theorem 5.1 provides the unique weak solvability of Problem ๐’ฌ\mathcal{Q}. Moreover, the regularity of the weak solution is ๐’–โˆˆC(โ„+;U),๐ˆโˆˆC(โ„+;Q1)\boldsymbol{u}\in C\left(\mathbb{R}_{+};U\right),\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q_{1}\right).

6 Penalization

In this section we show how the abstract result in Theorem 3.2 can be used in the study of the contact problem ๐’ฌ\mathcal{Q}. To this end, for each ฮผ>0\mu>0 we consider the following contact problem.

Problem ๐’ฌฮผ\mathcal{Q}_{\mu}. Find a displacement field ๐’–ฮผ:ฮฉร—โ„+โ†’โ„d\boldsymbol{u}_{\mu}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R}^{d} and a stress field ฯƒฮผ:ฮฉร—โ„+โ†’๐•Šd\sigma_{\mu}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{S}^{d} such that

๐ˆฮผ(t)=๐’œ๐œบ(๐’–ฮผ(t))+โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–ฮผ(s))๐‘‘s in ฮฉ\displaystyle\boldsymbol{\sigma}_{\mu}(t)=\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)+\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(s)\right)ds\quad\text{ in }\quad\Omega (6.1)
Divโก๐ˆฮผ(t)+๐’‡0(t)=๐ŸŽ in ฮฉ\displaystyle\operatorname{Div}\boldsymbol{\sigma}_{\mu}(t)+\boldsymbol{f}_{0}(t)=\mathbf{0}\quad\text{ in }\quad\Omega (6.2)
๐’–ฮผ(t)=๐ŸŽ on ฮ“1\displaystyle\boldsymbol{u}_{\mu}(t)=\mathbf{0}\quad\text{ on }\quad\Gamma_{1} (6.3)
๐ˆฮผ(t)๐‚=๐’‡2(t) on ฮ“2\displaystyle\boldsymbol{\sigma}_{\mu}(t)\boldsymbol{\nu}=\boldsymbol{f}_{2}(t)\quad\text{ on }\quad\Gamma_{2} (6.4)
๐ˆฮผฯ„(t)=๐ŸŽ on ฮ“3\displaystyle\boldsymbol{\sigma}_{\mu\tau}(t)=\mathbf{0}\quad\text{ on }\quad\Gamma_{3} (6.5)

for all tโˆˆโ„+t\in\mathbb{R}_{+}, and there exists ฮพฮผ:ฮฉร—โ„+โ†’โ„\xi_{\mu}:\Omega\times\mathbb{R}_{+}\rightarrow\mathbb{R} which satisfies

ฯƒฮผฮฝ(t)+p(uฮผฮฝ(t))+1ฮผp(uฮผฮฝโˆ’g)+ฮพฮผ(t)=00โ‰คฮพฮผ(t)โ‰คFฮพฮผ(t)=0 if uฮผฮฝ(t)<0ฮพฮผ(t)=F if uฮผฮฝ(t)>0} on ฮ“3\left.\begin{array}[]{l}\sigma_{\mu\nu}(t)+p\left(u_{\mu\nu}(t)\right)+\frac{1}{\mu}p\left(u_{\mu\nu}-g\right)+\xi_{\mu}(t)=0\\ 0\leq\xi_{\mu}(t)\leq F\\ \xi_{\mu}(t)=0\text{ if }u_{\mu\nu}(t)<0\\ \xi_{\mu}(t)=F\text{ if }u_{\mu\nu}(t)>0\end{array}\right\}\quad\text{ on }\quad\Gamma_{3}

for all tโˆˆโ„+t\in\mathbb{R}_{+}.
Here and below uฮผฮฝu_{\mu\nu} and ๐ˆฮผฯ„\boldsymbol{\sigma}_{\mu\tau} represent the normal and the tangential components of the functions ๐’–ฮผ\boldsymbol{u}_{\mu} and ๐ˆฮผ\boldsymbol{\sigma}_{\mu}, respectively. Note that the contact condition (6.6) can be obtained from the contact condition (2.6) in the limit when gโ†’โˆžg\rightarrow\infty. For this reason, its mechanical interpretation is similar to that of condition (2.6) and could be summarised as follows: when there is separation between the bodyโ€™s surface and the foundation then the normal stress vanishes; the penetration arises only if the normal stress reaches the critical value FF; when there is penetration the contact follows a normal compliance condition of the form (2.13). For this reason we refer to this
condition as to a multivalued normal compliance contact condition. It models the case when the foundation is assumed to have a rigid-elastic behaviour. Arguments similar to those used in [9, 20] show that ฮผ\mu can be interpreted as a deformability coefficient of the hard layer of the foundation.

Using notation (5.10)-(5.12) by similar arguments as those used in the case of Problem ๐’ฌ\mathcal{Q} we obtain the following variational formulation of Problem ๐’ฌฮผ\mathcal{Q}_{\mu}.

Problem ๐’ฌฮผV\mathcal{Q}_{\mu}^{V}. Find a displacement field ๐’–ฮผ:โ„+โ†’V\boldsymbol{u}_{\mu}:\mathbb{R}_{+}\rightarrow V such that, for all tโˆˆโ„+t\in\mathbb{R}_{+}, the inequality below holds:

(๐’œ๐œบ(๐’–ฮผ(t)),๐œบ(๐’—)โˆ’๐œบ(๐’–ฮผ(t)))Q+(โˆซ0tโ„ฌ(tโˆ’s)๐œบ(๐’–ฮผ(s))๐‘‘s,๐œบ(๐’—)โˆ’๐œบ(๐’–ฮผ(t)))Q\displaystyle\left(\mathcal{A}\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right),\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)\right)_{Q}+\left(\int_{0}^{t}\mathcal{B}(t-s)\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(s)\right)ds,\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{\mu}(t)\right)\right)_{Q} (6.7)
+(P๐’–ฮผ(t),๐’—โˆ’๐’–ฮผ(t))V+1ฮผโˆซฮ“3p(uฮผฮฝ(t)โˆ’g)(vฮฝโˆ’uฮผฮฝ(t))๐‘‘a\displaystyle+\left(P\boldsymbol{u}_{\mu}(t),\boldsymbol{v}-\boldsymbol{u}_{\mu}(t)\right)_{V}+\frac{1}{\mu}\int_{\Gamma_{3}}p\left(u_{\mu\nu}(t)-g\right)\left(v_{\nu}-u_{\mu\nu}(t)\right)da
+j(๐’—)โˆ’j(๐’–ฮผ(t))โ‰ฅ(๐’‡(t),๐’—โˆ’๐’–ฮผ(t))Vโˆ€๐’—โˆˆV\displaystyle+j(\boldsymbol{v})-j\left(\boldsymbol{u}_{\mu}(t)\right)\geq\left(\boldsymbol{f}(t),\boldsymbol{v}-\boldsymbol{u}_{\mu}(t)\right)_{V}\quad\forall\boldsymbol{v}\in V

We have the following existence, uniqueness and convergence result, which states the unique solvability of Problem ๐’ฌฮผV\mathcal{Q}_{\mu}^{V} and describes the behaviour of its solution as ฮผโ†’0\mu\rightarrow 0.

Theorem 6.1 Assume that (5.4)-(5.8) hold. Then:

  1. 1.

    For each ฮผ>0\mu>0 Problem ๐’ฌฮผV\mathcal{Q}_{\mu}^{V} has a unique solution which satisfies ๐’–ฮผโˆˆC(โ„+;V)\boldsymbol{u}_{\mu}\in C\left(\mathbb{R}_{+};V\right).

  2. 2.

    The solution ๐’–ฮผ\boldsymbol{u}_{\mu} of Problem ๐’ฌฮผV\mathcal{Q}_{\mu}^{V} converges to the solution ๐’–\boldsymbol{u} of Problem ๐’ฌV\mathcal{Q}^{V}, that is

โ€–๐’–ฮผ(t)โˆ’๐’–(t)โ€–Vโ†’0 as ฮผโ†’0\left\|\boldsymbol{u}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V}\rightarrow 0\quad\text{ as }\quad\mu\rightarrow 0 (6.8)

for all tโˆˆโ„+t\in\mathbb{R}_{+}.
Proof. We use Theorem 3.2 with X=VX=V and K=UK=U. To this end we define the operator G:Vโ†’VG:V\rightarrow V by equality

(G๐’–,๐’—)V=โˆซฮ“3p(uฮฝโˆ’g)vฮฝ๐‘‘aโˆ€๐’–,๐’—โˆˆV(G\boldsymbol{u},\boldsymbol{v})_{V}=\int_{\Gamma_{3}}p\left(u_{\nu}-g\right)v_{\nu}da\quad\forall\boldsymbol{u},\boldsymbol{v}\in V (6.9)

We use (5.1) and (5.7) to show that GG is a monotone Lipschitz continuous operator with Lipschitz constant M=c02LpM=c_{0}^{2}L_{p}, i.e. it satisfies condition (3.8) (a) and (b).

Assume now that ๐’–โˆˆV\boldsymbol{u}\in V and ๐’—โˆˆU\boldsymbol{v}\in U. Then, using (5.9) and (5.7) it is easy to see that

p(uฮฝโˆ’g)(vฮฝโˆ’g)โ‰ค0 a.e. on ฮ“3\displaystyle p\left(u_{\nu}-g\right)\left(v_{\nu}-g\right)\leq 0\quad\text{ a.e. on }\Gamma_{3}
p(uฮฝโˆ’g)(gโˆ’uฮฝ)โ‰ค0 a.e. on ฮ“3\displaystyle p\left(u_{\nu}-g\right)\left(g-u_{\nu}\right)\leq 0\quad\text{ a.e. on }\Gamma_{3}

and, therefore

(G๐’–,๐’—โˆ’๐’–)V=โˆซฮ“3p(uฮฝโˆ’g)(vฮฝโˆ’uฮฝ)๐‘‘a\displaystyle(G\boldsymbol{u},\boldsymbol{v}-\boldsymbol{u})_{V}=\int_{\Gamma_{3}}p\left(u_{\nu}-g\right)\left(v_{\nu}-u_{\nu}\right)da
=โˆซฮ“3p(uฮฝโˆ’g)(vฮฝโˆ’g)๐‘‘a+โˆซฮ“3p(uฮฝโˆ’g)(gโˆ’uฮฝ)๐‘‘aโ‰ค0\displaystyle\quad=\int_{\Gamma_{3}}p\left(u_{\nu}-g\right)\left(v_{\nu}-g\right)da+\int_{\Gamma_{3}}p\left(u_{\nu}-g\right)\left(g-u_{\nu}\right)da\leq 0

which shows that (3.8) (c) holds, too.
Finally, assume that G๐’–=๐ŸŽVG\boldsymbol{u}=\mathbf{0}_{V}. Then, (G๐’–,๐’–)V=0(G\boldsymbol{u},\boldsymbol{u})_{V}=0 and, therefore,

โˆซฮ“3p(uฮฝโˆ’g)uฮฝ๐‘‘a=0\int_{\Gamma_{3}}p\left(u_{\nu}-g\right)u_{\nu}da=0 (6.10)

We use (5.7) to obtain the inequality

p(uฮฝโˆ’g)uฮฝโ‰ฅp(uฮฝโˆ’g)gโ‰ฅ0 a.e. on ฮ“3p\left(u_{\nu}-g\right)u_{\nu}\geq p\left(u_{\nu}-g\right)g\geq 0\quad\text{ a.e. on }\Gamma_{3}

Therefore, since the integrand in (6.10) is positive, we deduce from (6.10) that

p(uฮฝโˆ’g)uฮฝ=0 a.e. on ฮ“3p\left(u_{\nu}-g\right)u_{\nu}=0\quad\text{ a.e. on }\Gamma_{3}

This equality combined with assumption (5.7) (d) implies that uฮฝโ‰คgu_{\nu}\leq g a.e. on ฮ“3\Gamma_{3} and, therefore, we deduce that ๐’–โˆˆU\boldsymbol{u}\in U. Conversely, if ๐’–โˆˆU\boldsymbol{u}\in U it follows that uฮฝโ‰คgu_{\nu}\leq g a.e. on ฮ“3\Gamma_{3} and using assumption (5.7) (d) we deduce that p(uฮฝโˆ’g)=0p\left(u_{\nu}-g\right)=0 a.e. on ฮ“3\Gamma_{3}. From the definition (6.9) of the operator GG we deduce that (G๐’–,๐’—)V=0(G\boldsymbol{u},\boldsymbol{v})_{V}=0 for all ๐’—โˆˆV\boldsymbol{v}\in V, which implies that G๐’–=0VG\boldsymbol{u}=0_{V}. It follows from above that GG satisfies the condition (3.8)(d).

We now turn back to (5.19) and (5.20). Thus, it is easy to see that ๐’–ฮผ\boldsymbol{u}_{\mu} is a solution to Problem ๐’ฌฮผV\mathcal{Q}_{\mu}^{V} iff

(A๐’–ฮผ(t)\displaystyle\left(A\boldsymbol{u}_{\mu}(t)\right. ,๐’—โˆ’๐’–ฮผ(t))V+(๐’ฎ๐’–ฮผ(t),๐’—โˆ’๐’–ฮผ(t))V+1ฮผ(G๐’–ฮผ(t),๐’—โˆ’๐’–ฮผ(t))V\displaystyle\left.,\boldsymbol{v}-\boldsymbol{u}_{\mu}(t)\right)_{V}+\left(\mathcal{S}\boldsymbol{u}_{\mu}(t),\boldsymbol{v}-\boldsymbol{u}_{\mu}(t)\right)_{V}+\frac{1}{\mu}\left(G\boldsymbol{u}_{\mu}(t),\boldsymbol{v}-\boldsymbol{u}_{\mu}(t)\right)_{V} (6.11)
+j(๐’—)โˆ’j(๐’–ฮผ(t))โ‰ฅ(๐’‡(t),๐’—โˆ’๐’–ฮผ(t))Vโˆ€๐’—โˆˆV\displaystyle+j(\boldsymbol{v})-j\left(\boldsymbol{u}_{\mu}(t)\right)\geq\left(\boldsymbol{f}(t),\boldsymbol{v}-\boldsymbol{u}_{\mu}(t)\right)_{V}\quad\forall\boldsymbol{v}\in V

for all tโˆˆโ„+t\in\mathbb{R}_{+}. Moreover, ๐’–\boldsymbol{u} is a solution to Problem ๐’ฌV\mathcal{Q}^{V} iff ๐’–\boldsymbol{u} satisfies inequality (5.23) for all tโˆˆโ„+t\in\mathbb{R}_{+}. Recall also that the operator GG satisfies condition (3.8). Theorem 6.1 is now a consequence of Theorem 3.2.

Note that the convergence result (6.8) can be easily extended to the weak solutions of the problems ๐’ฌฮผ\mathcal{Q}_{\mu} and ๐’ฌ\mathcal{Q}. Indeed, let ๐ˆฮผ\boldsymbol{\sigma}_{\mu} and ๐ˆ\boldsymbol{\sigma} be the functions defined by (6.1) and (2.1), respectively, and let tโˆˆโ„+,nโˆˆโ„•โˆ—t\in\mathbb{R}_{+},n\in\mathbb{N}^{*} be such that tโˆˆ[0,n]t\in[0,n]. Then, following the arguments presented in Section 5, it follows that ๐ˆฮผ,๐ˆโˆˆC(โ„+;Q)\boldsymbol{\sigma}_{\mu},\boldsymbol{\sigma}\in C\left(\mathbb{R}_{+};Q\right) and, moreover,

Divโก๐ˆฮผ(t)=Divโก๐ˆ(t)=โˆ’๐’‡0(t)\operatorname{Div}\boldsymbol{\sigma}_{\mu}(t)=\operatorname{Div}\boldsymbol{\sigma}(t)=-\boldsymbol{f}_{0}(t) (6.12)

Therefore, using (2.1), (6.1) and (6.12) as well as the properties of the operators ๐’œ\mathcal{A} and โ„ฌ\mathcal{B} we deduce that

โ€–๐ˆฮผ(t)โˆ’๐ˆ(t)โ€–Q1=โ€–๐ˆฮผ(t)โˆ’๐ˆ(t)โ€–Qโ‰คL๐’œโ€–๐’–ฮผ(t)โˆ’๐’–(t)โ€–V\displaystyle\left\|\boldsymbol{\sigma}_{\mu}(t)-\boldsymbol{\sigma}(t)\right\|_{Q_{1}}=\left\|\boldsymbol{\sigma}_{\mu}(t)-\boldsymbol{\sigma}(t)\right\|_{Q}\leq L_{\mathcal{A}}\left\|\boldsymbol{u}_{\mu}(t)-\boldsymbol{u}(t)\right\|_{V} (6.13)
+dmaxrโˆˆ[0,n]โกโ€–โ„ฌ(r)โ€–๐โˆžโˆซ0nโ€–๐’–ฮผ(s)โˆ’๐’–(s)โ€–V๐‘‘s\displaystyle\quad+d\max_{r\in[0,n]}\|\mathcal{B}(r)\|_{\mathbf{Q}_{\infty}}\int_{0}^{n}\left\|\boldsymbol{u}_{\mu}(s)-\boldsymbol{u}(s)\right\|_{V}ds

Next, we take ๐’—=๐ŸŽV\boldsymbol{v}=\mathbf{0}_{V} in (6.11), then we use the properties of the operators A,GA,G combined with those of the functional jj. As a result we obtain

m๐’œโ€–๐’–ฮผ(t)โ€–Vโ‰คโ€–A๐ŸŽVโ€–V+โ€–๐’ฎ๐’–ฮผ(t)โ€–V+โ€–๐’‡(t)โ€–V.m_{\mathcal{A}}\left\|\boldsymbol{u}_{\mu}(t)\right\|_{V}\leq\left\|A\mathbf{0}_{V}\right\|_{V}+\left\|\mathcal{S}\boldsymbol{u}_{\mu}(t)\right\|_{V}+\|\boldsymbol{f}(t)\|_{V}.

We now use the property (5.21) of the operator ๐’ฎ\mathcal{S} and a Gronwall argument to see that

โ€–๐’–ฮผ(t)โ€–Vโ‰คcn\left\|\boldsymbol{u}_{\mu}(t)\right\|_{V}\leq c_{n} (6.14)

where cnc_{n} represents a constant which depends on nn but is independent on ฮผ\mu. Then, we use the inequality (6.13), the convergence (6.8), the estimate (6.14) and Lebesqueโ€™s theorem to deduce that

โ€–๐ˆฮผ(t)โˆ’๐ˆ(t)โ€–Q1โ†’0 as ฮผโ†’0.\left\|\boldsymbol{\sigma}_{\mu}(t)-\boldsymbol{\sigma}(t)\right\|_{Q_{1}}\rightarrow 0\quad\text{ as }\quad\mu\rightarrow 0. (6.15)

In addition to the mathematical interest in the convergence result (6.8), (6.15), it is important from the mechanical point of view, since it shows that the weak solution of the viscoelastic contact problem with multivalued normal compliance and unilateral constraint may be approached as closely as one wishes by the solution of the viscoelastic contact problem with multivalued normal compliance, with a sufficiently small deformability coefficient.

7 Conclusion

We presented a penalization method for a class of history-dependent variational inequalities in Hilbert spaces. It contains the existence and the uniqueness of the solution for the penalized problems as well as its convergence to the solution of the original problem. The proofs were based on arguments of compactness and monotonicity. The method can be applied in the study of a large class of nonlinear boundary value problems with unilateral constraints. To provide an example, we presented a new model of quasistatic frictionless contact with viscoelastic materials which, in the variational formulation, leads to a history-dependent variational inequality for the displacement field. We applied the abstract penalization method in the study of this contact problem and we presented the mechanical interpretation of the corresponding results. A numerical validation of the convergence result included in this method will be provided in a forthcoming paper.

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