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Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions
Journal of Inequalities and Applications volume 2009, Article number: 207873 (2009)
Abstract
We prove the regularity for solutions of parabolic hemivariational inequalities of dynamic elasticity in the strong sense and investigate the continuity of the solution mapping from initial data and forcing term to trajectories.
1. Introduction
In this paper, we deal with the existence and a variational of constant formula for solutions of a parabolic hemivariational inequality of the form:
where is a bounded domain in with sufficiently smooth boundary Let , , The boundary is composed of two pieces and , which are nonempty sets and defined by
where is the unit outward normal vector to . Here , is the displacement, is the strain tensor, , is a multivalued mapping by filling in jumps of a locally bounded function , . A continuous map from the space of symmetric matrices into itself is defined by
where is the identity of , denotes the trace of , and , . For example, in the case , , where is Young's modulus, is Poisson's ratio and is the density of the plate.
Let and be two complex Hilbert spaces. Assume that is a dense subspace in and the injection of into is continuous. Let be a continuous linear operator from into which is assumed to satisfy Gårding's inequality. Namely, we formulated the problem (1.1) as
The existence of global weak solutions for a class of hemivariational inequalities has been studied by many authors, for example, parabolic type problems in [1–4], and hyperbolic types in [5–7]. Rauch [8] and Miettinen and Panagiotopoulos [1, 2] proved the existence of weak solutions for elliptic one. The background of these variational problems are physics, especially in solid mechanics, where nonconvex and multivalued constitutive laws lead to differential inclusions. We refer to [3, 4] to see the applications of differential inclusions. Most of them considered the existence of weak solutions for differential inclusions of various forms by using the FaedoGalerkin approximation method.
In this paper, we prove the existence and a variational of constant formula for strong solutions of parabolic hemivariational inequalities. The plan of this paper is as follows. In Section 2, the main results besides notations and assumptions are stated. In order to prove the solvability of the linear case with we establish necessary estimates applying the result of Di Blasio et al. [9] to (1.1)–(1.5) considered as an equation in as well as . The existence and regularity for the nondegenerate nonlinear systems has been developed as seen in [10, Theorem 4.1] or [11, Theorem 2.6], and the references therein. In Section 3, we will obtain the existence for solutions of (1.1)–(1.5) by converting the problem into the contraction mapping principle and the norm estimate of a solution of the above nonlinear equation on . Consequently, if is a solution asociated with , and , in view of the monotonicity of , we show that the mapping
is continuous.
2. Preliminaries and Linear Hemivariational Inequalities
We denote for , and for . Throughout this paper, we consider
We denote the dual space of , the dual pairing between and .
The norms on , , and will be denoted by , and , respectively. For the sake of simplicity, we may consider
We denote by . Let be the operator associated with a sesquilinear form which is defined Gårding's inequality
that is,
Then is a symmetric bounded linear operator from into which satisfies
and its realization in which is the restriction of to
is also denoted by . Here, we note that is dense in . Hence, it is also dense in . We endow the domain of with graph norm, that is, for , we define . So, for the brevity, we may regard that for all . It is known that— generates an analytic semigroup in both and .
From the following inequalities
it follows that there exists a constant such that
So, we may regard as where is the real interpolation space between and .
Consider the following initial value problem for the abstract linear parabolic type equation:
A continuous map from the space of symmetric matrices into itself is defined by
It is easily known that
Note that the map is linear and symmetric and it can be easily verified that the tensor satisfies the condition
Let be the smallest positive constant such that
Simple calculations and Korn's inequality yield that
and hence is equivalent to the norm on Then by virtue of [9, Theorem 3.3], we have the following result on the linear parabolic type equation (LE).
Proposition 2.1.
Suppose that the assumptions stated above are satisfied. Then the following properties hold.
(1)For any and , there exists a unique solution of (LE) belonging to
and satisfying
where is a constant depending on .
(2)Let and for any . Then there exists a unique solution of (LE) belonging to
and satisfying
where is a constant depending on .
Proof.

(1)
Let be a bounded sesquilinear form defined in by
(2.19)
Noting that by(2.10)
and by (2.12), (2.14), and (1.6),
it follows that there exist and such that
Let be the operator associated with this sesquilinear form:
Then is also a symmetric continuous linear operator from into which satisfies
So we know that— generates an analytic semigroup in both and . Hence, by applying [9, Theorem 3.3] to the regularity for the solution of the equation:
in the space , we can obtain a unique solution of (LE) belonging to
and satisfying the norm estimate (2.16).

(2)
It is easily seen that
(2.27)
for the time . Therefore, in terms of the intermediate theory we can see that
and follow the argument of (1) term by term to deduce the proof of (2) results.
3. Existence of Solutions in the Strong Sense
This Section is to investigate the regularity of solutions for the following parabolic hemivariational inequality of dynamic elasticity in the strong sense:
Now, we formulate the following assumptions.
(Hb) Let be a locally bounded function verifying
where We denote
The multivalued function is obtained by filling in jumps of a function by means of the functions as follows.
We denote , for . We will need a regularization of defined by
where and It is easy to show that is continuous for all and satisfy the same condition (Hb) with possibly different constants if satisfies (Hb). It is also known that is locally Lipschitz continuous in , that is for any , there exists a number such that
(Hb1)
holds for all with We denote
The following lemma is from [[12]; Lemma A.5].
Lemma 3.1.
Let satisfying for all and be a constant. Let be a continuous function on satisfying the following inequality:
Then,
Proof.
Let
Then
and
Hence, we have
Since is absolutely continuous and
for all , it holds
that is,
Therefore, combining this with (3.11), we conclude that
for arbitrary .
From now on, we establish the following results on the local solvability of the following equation,
Lemma 3.2.
Let be a solution of (HIE1) and . Then, the following inequality holds, for any ,
where .
Proof.
We remark that from (2.11), (2.12), it follows that there is a constant such that
Consider the following equation:
Multipying on both sides of , we get
and integrating this over , by (1.6), (2.5), (3.18) and (Hb1), we have
that is,
Applying Gronwall lemma, the proof of the lemma is complete.
Theorem 3.3.
Assume that , and (Hb). Then, there exists a time such that (HIE1) admits a unique solution
Proof.
Assume that (2.5) holds for . Let the constant satisfy the following inequality:
Let us fix such that
where is given by (Hb).
Invoking Proposition 2.1, for a given , the problem
has a unique solution . To prove the existence and uniqueness of solutions of semilinear type (HIE1), by virtue of Lemma 3.2, we are going to show that the mapping defined by maps is strictly contractive from into itself if the condition (3.25) is satisfied.
Lemma 3.4.
Let be the solutions of (HIE2) with replaced by where is the ball of radius centered at zero of , respectively. Then the following inequality holds:
where
Proof.
Let be the solutions of (HIE2) with replaced by , respectively. Then, we have that
Multiplying on both sides of and by (2.8), we get
and so, by (3.18), (2.5), (Hb), we obtain
Putting
and integrating (3.30) over , this yields
From (3.32) it follows that
Integrating (3.33) over we have
thus, we get
From (3.32) and (3.35) it follows that
which implies
By using Lemma 3.1, we obtain that
The proof of lemma is complete.
From (3.26) and (3.36) it follows that
Starting from the initial value , consider a sequence satisfying
Then from (3.39) it follows that
So by virtue of the condition (3.25) the contraction principle gives that there exists such that
and hence, from (3.41) there exists such that
Now, we give a norm estimation of the solution (HIE) and establish the global existence of solutions with the aid of norm estimations.
Theorem 3.5.
Let the assumption (Hb) be satisfied. Assume that and for any . Then, the solution of (HIE) exists and is unique in
Furthermore, there exists a constant depending on such that
Proof.
Let be the solution of
Then, since
by multiplying by , from (Hb), (3.18) and the monotonicity of , we obtain
By integrating on (3.48) over we have
By the procedure similar to (3.39) we have
Put
Then it holds
and hence, from (2.16) in Proposition 2.1, we have that
for some positive constant . Noting that by (Hb)
and by Proposition 2.1
it is easy to obtain the norm estimate of in satisfying (3.45).
Now from Theorem 3.3 it follows that
So, we can solve the equation in and obtain an analogous estimate to (3.53). Since the condition (3.25) is independent of initial values, the solution of (HIE1) can be extended the internal for a natural number , that is, for the initial in the interval , as analogous estimate (3.53) holds for the solution in . Furthermore, the estimate (3.45) is easily obtained from (3.53) and (3.56).
We show that is a solution of the problem (HIE). Lemma 3.4 and (Hb) give that
and for , there exists a unique solution of (HIE) belonging to
and satisfying (3.44).
From (3.44)and (3.57), we can extract a subsequence from , still denoted by , such that
Here, we remark that if is compactly embedded in and , the following embedding
is compact in view [13, Theorem 2]. Hence, the mapping
By a solution of (HIE1), we understand a mild solution that has a form
so letting and using the convergence results above, we obtain
Now, we show that a.e. in . Indeed, from (3.62) we have strongly in and hence a.e. in for each Let and Using the theorem of Lusin and Egoroff, we can choose a subset such that and uniformly on Thus, for each there is an such that
Then, if we have for all and
Therefore we have
Let Then
Letting in this inequality and using (3.60), we obtain
where Letting in this inequality, we deduce that
and letting we get
This implies that a.e. in This completes the proof of theorem.
Remark 3.6.
In terms of Proposition 2.1, we remark that if and for any then the solution of (HIE) exists and is unique in
Futhermore, there exists a constant depending on such that
Theorem 3.7.
Let the assumption (Hb) be satisfied
(1)if , then the solution of (HIE) belongs to and the mapping
is continuous.

(2)
let . Then the solution of (HIE) belongs to and the mapping
(3.74)
is continuous.
Proof.

(1)
It is easy to show that if and , then belongs to . let and be the solution of (HIE) with in place of for Then in view of Proposition 2.1, we have
(3.75)
Since
we get
Hence, arguing as in (2.8), we get
Combining (3.75) with (3.78), we obtain
Suppose that in and let and be the solutions (HIE) with and , respectively. Let be such that
Then by virtue of (3.79) with replaced by we see that
This implies that in . Hence the same argument shows that in
Repeating this process we conclude that in

(2)
If then belongs to from Theorem 3.5. Let and be the solution of (HIE) with in place of for . Multiplying (HIE) by , we have
(3.83)
Put
Then, by the similar argument in (3.32), we get
and we have that
thus, arguing as in (3.35) we have
Combining this inequality with (3.85) it holds that
By Lemma 3.1 the following inequality
implies that
Hence, from (3.88) and (3.90) it follows that
The last term of (3.91) is estimated as
Let be such that
Hence, from (3.91) and (3.92) it follows that there exists a constant such that
Suppose in , and let and be the solutions (HIE) with and , respectively. Then, by virtue of (3.94), we see that in . This implies that in . Therefore the same argument shows that in
Repeating this process, we conclude that in .
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The authors wish to thank the referees for careful reading of manuscript, for valuable suggestions and many useful comments.
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Park, DG., Jeong, JM. & Park, S.H. Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions. J Inequal Appl 2009, 207873 (2009). https://doi.org/10.1155/2009/207873
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DOI: https://doi.org/10.1155/2009/207873
Keywords
 Weak Solution
 Mild Solution
 Differential Inclusion
 Continuous Linear Operator
 Analytic Semigroup
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