# Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions

- Dong-Gun Park
^{1}, - Jin-Mun Jeong
^{2}Email author and - Sun Hye Park
^{3}

**2009**:207873

**DOI: **10.1155/2009/207873

© Dong-Gun Park et al. 2009

**Received: **28 August 2008

**Accepted: **1 January 2009

**Published: **5 January 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

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 [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 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.

is continuous.

## 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).

Proposition 2.1.

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 [[12]; Lemma A.5].

Lemma 3.1.

Proof.

for arbitrary .

Lemma 3.2.

where .

Proof.

Applying Gronwall lemma, the proof of the lemma is complete.

Theorem 3.3.

Proof.

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.

Lemma 3.4.

Proof.

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.

Theorem 3.5.

Proof.

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.

Remark 3.6.

Theorem 3.7.

Let the assumption (Hb) be satisfied

- (2)let . Then the solution of (HIE) belongs to and the mapping(3.74)

is continuous.

- (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 .

## Declarations

### Acknowledgment

The authors wish to thank the referees for careful reading of manuscript, for valuable suggestions and many useful comments.

## Authors’ Affiliations

## References

- Miettinen M:
**A parabolic hemivariational inequality.***Nonlinear Analysis: Theory, Methods & Applications*1996,**26**(4):725–734. 10.1016/0362-546X(94)00312-6MathSciNetView ArticleMATHGoogle Scholar - Miettinen M, Panagiotopoulos PD:
**On parabolic hemivariational inequalities and applications.***Nonlinear Analysis: Theory, Methods & Applications*1999,**35**(7):885–915. 10.1016/S0362-546X(97)00720-7MathSciNetView ArticleMATHGoogle Scholar - Panagiotopoulos PD:
*Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions*. Birkhäuser, Boston, Mass, USA; 1985:xx+412.View ArticleGoogle Scholar - Panagiotopoulos PD:
**Modelling of nonconvex nonsmooth energy problems. Dynamic hemivariational inequalities with impact effects.***Journal of Computational and Applied Mathematics*1995,**63**(1–3):123–138.MathSciNetView ArticleMATHGoogle Scholar - Park JY, Kim HM, Park SH:
**On weak solutions for hyperbolic differential inclusion with discontinuous nonlinearities.***Nonlinear Analysis: Theory, Methods & Applications*2003,**55**(1–2):103–113. 10.1016/S0362-546X(03)00216-5View ArticleMathSciNetMATHGoogle Scholar - Park JY, Park SH:
**On solutions for a hyperbolic system with differential inclusion and memory source term on the boundary.***Nonlinear Analysis: Theory, Methods & Applications*2004,**57**(3):459–472. 10.1016/j.na.2004.02.024MathSciNetView ArticleMATHGoogle Scholar - Migórski S, Ochal A:
**Vanishing viscosity for hemivariational inequalities modeling dynamic problems in elasticity.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(8):1840–1852. 10.1016/j.na.2006.02.028MathSciNetView ArticleMATHGoogle Scholar - Rauch J:
**Discontinuous semilinear differential equations and multiple valued maps.***Proceedings of the American Mathematical Society*1977,**64**(2):277–282. 10.1090/S0002-9939-1977-0442453-6MathSciNetView ArticleMATHGoogle Scholar - Di Blasio G, Kunisch K, Sinestrari E:
**-regularity for parabolic partial integro-differential equations with delay in the highest-order derivatives.***Journal of Mathematical Analysis and Applications*1984,**102**(1):38–57. 10.1016/0022-247X(84)90200-2MathSciNetView ArticleMATHGoogle Scholar - Ahmed NU:
*Optimization and Identification of Systems Governed by Evolution Equations on Banach Space, Pitman Research Notes in Mathematics Series*.*Volume 184*. Longman Scientific & Technical, Harlow, UK; 1988:vi+187.Google Scholar - Barbu V:
*Nonlinear Semigroups and Differential Equations in Banach Spaces*. Nordhoff, Leiden, The Netherlands; 1976:352.View ArticleGoogle Scholar - Brézis H:
*Opérateurs Maximaux Monotones et Semigroupes de Contractions dans un Espace de Hilbert*. North Holland, Amsterdam, The Netherlands; 1973.MATHGoogle Scholar - Aubin J-P:
**Un théorème de compacité.***Comptes Rendus de l'Académie des Sciences*1963,**256:**5042–5044.MathSciNetMATHGoogle Scholar

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