Semilinear problems with poly-Laplace type operators
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Semilinear problems with poly-Laplace type operators
Radu Precup
Faculty of Mathematics and Computer Science and Institute of
Advanced Studies in Science and Technology,
Babeş-Bolyai University, 400084 Cluj-Napoca, Romania &
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian
Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania
Abstract
The paper deals with semilinear operator equations involving iterates of a
strongly monotone symmetric linear operator. In particular there are
consider semilinear polyharmonic equations subject to the Navier boundary
conditions. A careful analysis is made on the energetic spaces associated to
such problems and a number of existence results are obtained by using a
fixed point approach.
Mathematics Classification: 35J40, 46E35, 47J25
Key words: Polyharmonic equation, iterates of symmetric linear operators,
energetic space.
1 Introduction
There is known the bi-Laplace equation whose solutions
are called biharmonic functions. The equation arises as a model for
the elastic equilibrium in the theory of elasticity. Also there are known
its generalizations, the poly-Laplace equations
whose solutions are said to be polyharmonic of order (see, [9] and [10]). The operator is
referred as the bi-Laplacian and is said to be the Laplacian of order
The non-homogeneous versions of these equations are
and when considered in a domain in case there has been
added the boundary condition
where is the unit normal vector to the boundary, or the boundary
condition
(see [8]). For condition (1.1) can be generalized following
Lauricella [7] as follows
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and (1.2), as suggested by Riquier [13], by
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For the classical theory of polyharmonic functions we refer the reader to
the volume [9] which brings together the entire contribution of Miron
Nicolescu to this field and which allows obtaining information on
contributions originating from old, less accessible publications.
Modern theory has introduced the concept of weak solution and Sobolev spaces
as natural framework for the study of these operators and of the associated
semilinear problems. Thus the problem
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(1.3) |
involving the natural boundary condition can be naturally addressed in endowed
with the equivalent norm Other studies (see, e.g., [1], [2], [4]
and [11]) have aimed to treat problems of type (1.3) under the
boundary conditions on (called Navier
boundary conditions [5]) by looking for solutions in the space with norm The problem
is that the condition ”on ” not being a
natural boundary condition it does not follow from the variational
formulation of the problem. This is the reason to restrict the study to a
subspace of functions in order to give a meaning to the equality
on the boundary. This will be one of our main goals in this work. Roughly
speaking we suggest that the iterative nature of the differential operator
to reflect on its energetic space and consequently on some basic
inequalities. We lead this discussion more generally considering instead of
Laplacian a strongly monotone symmetric linear operator Thus our
results will concern semilinear operator equations of the form
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where is the -th iterate of defined recursively by The whole approach is based on the theory of the energetic
space associated to There are thus obtained existence results
for the problem
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(1.4) |
where is given and is on the position of a
perturbation of In particular, we obtain results for semilinear
poly-Laplace equations.
2 Preliminaries
In this section we recall the notion of energetic space (see [14]) and
some related results.
2.1 The energetic space
Let be a real Hilbert space with the inner product and the norm Let be a
linear subspace of and be a strongly monotone
symmetric linear operator, that is, a linear operator satisfying
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(2.1) |
and some constant Then, endowed with the energetic inner
product
and the energetic norm
becomes a pre-Hilbert space. Its completion (see, e.g., [6, Section I
4.3]) denoted by is called the energetic space of
In virtue of (2.1), any Cauchy sequence in the energetic norm is also a
Cauchy sequence in the norm This allows
us to see as a subset of the original complete space and the
elements of as limits in of Cauchy sequences from Furthemore, the energetic inner
product and norm can be extended from to by
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where and are Cauchy
sequences in that
converge in to and respectively.
2.2 Abstract Poincaré’s inequality
Inequality (2.1) can be extended by density from to showing
that
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(2.2) |
Thus is an embedding constant for the continuous inclusion We call this inequality Poincaré’s inequality.
If the embedding is compact, then there is a largest
embedding constant and the inequality is reached. Indeed, if we denote
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then and if we take any minimizing sequence , that is
then using the compactness of the embedding and passing
eventually to a subsequence we can assume that in
for some Furthermore, from the identity
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since we deduce
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as Hence is a Cauchy
sequence in Let be such that in Then in too, and the uniqueness of the
limit implies that Consequently, that is Thus the infimum is
reached and is the best constant in (2.2). Thus,
in case that the embedding is compact, Poincaré’s
inequality reads as follows:
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2.3 The dual of the energetic space
Having for the dual spaces we have and if, based on Riesz’ theorem, we assume the
identification then one has
In addition, from (2.2) we also have
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(2.3) |
Indeed, if then for any one has where by we mean the value of a linear
functional at a given element. Then
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Notice in case that the embedding is compact, so is the
embedding and in (2.3) we may take as in (2.2) the best constant
2.4 Extension of operator
Clearly we can define the linear operator by
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In particular, if then since one has which by the density of into can be
extended to all Thus the functionals and act identically in The last one is a
continuous linear functional on which in virtue of Riesz’s
representation theorem is identified with In this sense, as continuous
linear functionals on one has and therefore can be seen as an extension of from to It is
common to use the same symbol for the extension Thus
2.5 The inverse of operator
In the previous subsection we have that for every there is a
unique element denoted with
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(2.4) |
Conversely, for every by Riesz’s theorem, there is a
unique element with
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Clearly and thus Hence the inverse of is the
operator defined by
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The two linear operators and are isometries between and
Indeed, letting in (2.4) gives
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whence The converse inequality comes from
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Hence
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2.6 Weak solutions to linear operator equations
Consider the operator equation associated to
By a (strong) solution we mean an element such that
Obviously this is possible if By a weak solution we mean
an element satisfying the identity
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When speaking about weak solutions we may assume more generally that In view of the previous subsection, for each the equation has a unique weak solution, namely
Note that looking for weak solutions to a semilinear equation
where is any mapping, reduces to
solving the fixed point equation
3 Semilinear operator equations involving iterates of a symmetric
linear operator
We now come back to problem (1.4), where is a linear operator as in
Introduction.
3.1
Functional framework
Looking at the required conditions on the elements to belong to the energetic space of the operator we
may seek solutions in the space
Here for Since and one has
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(3.1) |
We endow with the inner product and norm
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Note that the functional is indeed a
norm on since if for some one has then whence and so on until we obtain
Lemma 3.1
The space endowed with the inner product is a Hilbert space which continuously embeds in
Proof. Let be any Cauchy sequence in Then is Cauchy in so convergent in to some
Since the embedding is
continuous, we then have in
Next, the continuity of from to implies in Repeating the above reasoning
we arrive to the conclusion that
in that is in
This proves that is
complete.
Knowing the operator from to and the
inclusions (3.1) we immediately can see that for every there is a unique namely which
solves the non-homogeneous equation
Consequently, solving a semi-linear equation of the form
where is any mapping, is equivalent to
the fixed point equation
for the operator
3.2 Existence and uniqueness under a Lipschitz condition
Using Banach contraction principle we obtain the following result on problem
(1.4).
Theorem 3.2
Let satisfy
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(3.2) |
for all and some nonnegative constants If
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(3.3) |
then problem (1.4) has a unique solution
Proof. Problem (1.4) is equivalent to the fixed point equation
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Using (3.2) and Poincaré’s inequality (2.2), for any we have
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Furthermore, for ant one has
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Repeating the above estimations for we obtain
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(3.5) |
Then
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which in view of (3.3) shows that the operator
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is a contraction on The conclusion now follows from Banach contraction
principle.
3.3 Existence under a linear growth condition
If instead of the Lipschitz condition (3.2) we only have a linear
growth condition on and we assume that the embedding is
compact, then we can still prove the existence of at least one solution by
using Schauder’s fixed point theorem.
Theorem 3.3
Assume that the embedding is compact and that is continuous and satisfies
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(3.7) |
for all and some and
nonnegative constants If condition (3.3)
holds, then problem (1.4) has at least one solution with
Proof. As above we now have
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Since and is a self mapping of the
closed ball of centered at the origin and of radius On the
other hand can be decomposed as
where
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All these operators are continuous and bounded (send bounded sets to bounded
sets) and is compact. As a result their composition is
completely continuous. Now the conclusion follows from Schauder’s fixed
point theorem applied to in the ball
3.4 Existence via a priori bounds
We may replace the growth condition on by a sign type condition as shows
the following theorem.
Theorem 3.4
Assume that is
completely continuous and satisfies
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(3.8) |
for all and some Then problem (1.4) has at least one solution . Moreover, any solution of the problem satisfies
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(3.9) |
Proof. Using a similar reasoning as in the previous proof we can show that the
operator is completely continuous. We now prove
that the set of all possible solutions of the equations
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for is bounded as (3.9) shows. Indeed, if
is such a solution, then
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Next since based on (3.8), one has
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It follows that
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whence (3.9). The existence of a solution is guaranteed by the
Leray-Schauder principle.
4 Semilinear problems with poly-Laplace operators
The results established in Section 3 can be easily applied to problems
involving poly-Laplace operators, more exactly to the problem
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(4.1) |
Here is bounded open, and Hence
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Also the compactness of the imbedding holds and the
imbedding constant in Poincaré’s inequality is
where is the first eigenvalue of the Dirichlet problem for
Laplacian (for the theory of elliptic problems, see, e.g., [3] or [12]).
In this case, is the superposition operator
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Theorem 3.2 yields the following result.
Corollary 4.1
Let satisfy the Carathéodory conditions, and
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(4.2) |
for all and some nonnegative constants If ( being given by (3.3) with ),
then problem (4.1) has a unique solution
Proof. According to the main theorem abut Nemytskii’s superposition operator,
maps to In addition, for any from (4.2) we find
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Thus Theorem 3.2 is applicable and gives the result.
Theorem 3.3 yields the following result.
Corollary 4.2
Let satisfy the Carathéodory conditions and
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(4.3) |
for all a.e. some nonnegative constants and a function If ( being given by (3.3) with ), then
problem (4.1) has at least one solution with
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Using Theorem 3.4 we obtain the following result.
Corollary 4.3
Let satisfy the Carathéodory conditions and
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(4.4) |
for all a.e. some nonnegative constants a number
and a function where In addition assume that
for every and some Then problem (4.1) has at least one solution Moreover, any solution
satisfies
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Proof. Let Clearly Hence the
embedding is continuous, while since the embedding is compact. In addition since from (4.4) we have that Nemytskii’s superposition operator is
well-defined, continuous and bounded from to Then our operator can be decomposed
as where
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Since is compact one deduces that is completely continuous.
We now check condition (3.8). For one has
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Hence the assumptions of Theorem 3.4 are fulfilled and the conclusion
follows.
In contrast with the general case of equations involving iterates of a
linear operator the case of the Laplace operator is a special one due
to the representation of the Laplacian as
a composition of two differential operators, the gradient and the
divergence. This particularity allows nonlinear terms of semilinear
equations also to depend on gradient. Thus, instead of problem (4.1) we
can consider more generally the problem
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(4.5) |
Then looking to extend to this problem the results in Corollaries 4.1
and 4.2, the expression of constant in (3.3) should be
completed by terms involving odd powers of ().
For example, if is such that
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for all and (where applied to vectors from notation stands for the Euclidian
norm), then trying to follow the estimation made for (3.2) we arrive to
the final sum
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According to (3.5) we have
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which help in the estimation
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Then the analogue of (3.2) for the new operator
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is the estimate
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Thus the contraction condition guaranteeing the existence and uniqueness of
the solution of (4.5) is now
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An analogue result to Corollary 4.2 can be established under the growth
condition on
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and the same condition on the constants and
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