Boundary Stabilization of Hyperbolic Hemivariational Inequalities
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Boundary Stabilization of Hyperbolic Hemivariational Inequalities Sun Hye Park · Jong Yeoul Park · Jin Mun Jeong
Received: 2 April 2008 / Accepted: 19 May 2008 / Published online: 30 May 2008 © Springer Science+Business Media B.V. 2008
Abstract In this paper, we prove the existence of weak solutions and investigate uniform decay rates of global weak solutions for a hyperbolic hemivariational inequalitiy of dynamic elasticity. Keywords Hemivariational inequality · Existence of solution · Boundary stabilization · Decay rates Mathematics Subject Classification (2000) 35L85 · 35Q72 · 49J53
1 Introduction We are concerned with the global existence and the asymptotic stability of weak solutions for a hyperbolic hemivariational inequality with boundary damping: u¨ − div C[ε(u)] + = 0 in × (0, ∞), u=0
on 1 × (0, ∞),
C[ε(u)]ν = −(β · ν)u˙
(1.1) (1.2)
on 0 × (0, ∞),
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-355-C00002). S.H. Park Basic Sciences Research Institute, Pukyong National University, Busan 608-737, South Korea e-mail: [email protected] J.Y. Park () Department of Mathematics, Pusan National University, Busan 609-735, South Korea e-mail: [email protected] J.M. Jeong Division of Mathematical Sciences, Pukyong National University, Busan 608-737, South Korea e-mail: [email protected]
(1.3)
140
S.H. Park et al.
(x, t) ∈ ϕ(u(x, t)) a.e. (x, t) ∈ × (0, ∞), u(0) = u0 ,
u(0) ˙ = u1
in ,
(1.4) (1.5)
where be a bounded domain in RN with sufficiently smooth boundary . This inequality is considered on a Sobolev space of vector valued functions. Let x 0 ∈ RN , β(x) = x − x 0 , R = maxx∈ |x − x 0 |. The boundary is composed of two pieces 0 and 1 , which are nonempty sets and defined by 0 := {x ∈ : β(x) · ν ≥ α > 0} and
1 := {x ∈ : β(x) · ν ≤ 0}, 2
, u¨ = ∂∂t 2u , u = (u1 , . . . , uN )T where ν is the unit outward normal vector to . Here u˙ = ∂u ∂t 1 is the displacement, ε(u) = 2 {∇u + (∇u)T } is the strain tensor, ϕ(u) = (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] = a(tr ε)I + bε for ε ∈ S, 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, E C[ε] = d(1−μ 2 ) [μ(tr ε)I + (1 − μ)ε], where E > 0 is Young’s modulus, 0 < μ < 1/2 is Poisson’s ratio and d is the density of the plate. Note that the map C is linear and symmetric and it can be easily verified that the tensor C satisfies the condition λ0 |ε|2 ≤ C[ε] · ε ≤ λ1 |ε|2 , ε ∈ S
for some λ0 , λ1 > 0.
(1.6)
Recently, a class of hemivariational inequalities are studied by many authors [7–11, 13, 14]. Most of them considered the existence of weak solutions for differential inclusions of various forms. Rauch [14] and Miettinen and Panagiotopoulos [7, 8] proved the existence of weak solutions for
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