Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions
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Research Article Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions Dong-Gun Park,1 Jin-Mun Jeong,2 and Sun Hye Park3 1
Mathematics and Materials Physics, Dong-A University, Saha-Gu, Busan 604-714, South Korea Division of Mathematical Sciences, Pukyong National University, Busan 608-737, South Korea 3 Department of Mathematics, Pusan National University, Busan 609-735, South Korea 2
Correspondence should be addressed to Jin-Mun Jeong, [email protected] Received 28 August 2008; Revised 4 December 2008; Accepted 1 January 2009 Recommended by Donal O’Regan 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. Copyright q 2009 Dong-Gun Park et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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: ux, ˙ t Δux, t − div C εux, t Ξx, t fx, t
in Ω × 0, ∞,
ux, t 0 on Γ1 × 0, ∞, C ε ux, t ν −β · νux, t on Γ0 × 0, ∞, Ξx, t ∈ ϕ ux, t a.e. x, t ∈ Ω × 0, ∞, ux, 0 u0 x
in Ω,
1.1 1.2 1.3 1.4 1.5
where Ω is a bounded domain in RN with sufficiently smooth boundary Γ. Let x0 ∈ RN , βx x − x0 , R maxx∈Ω |x − x0 |. The boundary Γ is composed of two pieces Γ0 and Γ1 , which
2
Journal of Inequalities and Applications
are nonempty sets and defined by Γ0 : x ∈ Γ : βx · ν ≥ α > 0 ,
Γ1 : x ∈ Γ : βx · ν ≤ 0 ,
1.6
where ν is the unit outward normal vector to Γ. Here u˙ ∂u/∂t, u u1 , . . . , uN T is the displacement, εu 1/2{∇u ∇uT } 1/2∂ui /∂xj ∂uj /∂xi is the strain tensor, ϕu ϕ1 u1 , . . . , ϕN uN T , ϕi is a multi-valued mapping by filling in jumps of a locally bounded function bi , i 1, . . . , N. A continuous map C from the space S of N × N symmetric matrices into itself is defined by C ε atr εI bε,
for ε ∈ S,
1.7
where I is the identity of S, tr ε denotes the trace of ε, and a > 0, b > 0. For example, in the case N 2, C ε E/d1 − μ2 μtr εI 1 − με , where E > 0 is Young’s modulus, 0 < μ < 1/2 is Poisson’s ratio and d is the density of the plate. Let H and V be two complex Hilbert spaces. Assume that V is a dense subspace in H and the injection of V into H is continuous. Let A be a continuous linear operator from V into V ∗ which is assumed to satisfy G˚arding’s inequality. Namely, we formulated the problem 1.1 as u˙ Au − div C εu Ξ f
in Ω × 0, ∞.
1.8
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 hyperboli
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