Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions
© Dong-Gun Park et al. 2009
Received: 28 August 2008
Accepted: 1 January 2009
Published: 5 January 2009
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.
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.
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  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 multi-valued 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 Faedo-Galerkin approximation method.
2. Preliminaries and Linear Hemivariational Inequalities
We denote the dual space of , the dual pairing between and .
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 .
So, we may regard as where is the real interpolation space between and .
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).
Suppose that the assumptions stated above are satisfied. Then the following properties hold.
where is a constant depending on .
where is a constant depending on .
- (1)Let be a bounded sesquilinear form defined in by(2.19)
- (2)It is easily seen that(2.27)
and follow the argument of (1) term by term to deduce the proof of (2) results.
3. Existence of Solutions in the Strong Sense
Now, we formulate the following assumptions.
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
The following lemma is from [; Lemma A.5].
for arbitrary .
Applying Gronwall lemma, the proof of the lemma is complete.
where is given by (Hb).
has a unique solution . To prove the existence and uniqueness of solutions of semilinear type (HIE-1), 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.
The proof of lemma is complete.
Now, we give a norm estimation of the solution (HIE) and establish the global existence of solutions with the aid of norm estimations.
it is easy to obtain the norm estimate of in satisfying (3.45).
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 (HIE-1) 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).
and satisfying (3.44).
Then, if we have for all and
This implies that a.e. in This completes the proof of theorem.
Let the assumption (Hb) be satisfied
- (2)let . Then the solution of (HIE) belongs to and the mapping(3.74)
- (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)
- (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)
Repeating this process, we conclude that in .
The authors wish to thank the referees for careful reading of manuscript, for valuable suggestions and many useful comments.
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