A boundary thin obstacle problem for a wave equation

Abstract

In this paper we shall discuss a hyperbolic variational inequality associated with the following initial-boundary value problem:Here n is a bounded open subset of Rn, n ;:::: 2, with smooth boundary an E (0-1) (0-2) (0-3) (0-4) (0-5) C2.We assume that rl and r2 are disjoint, nonempty open subsets of an such that an = rl u r2 = rl u r2 and ar1 = ar2 E C 2 .The above functions J(x,t),uo(x),u1 (x) and 4>(z) are given, and f., denotes the outward normal derivative on an.We assume that 4> $ 0 and Uo ;:::: 4> on f 2 • We can put the above problem in the form of a variational iUP.quality: find u(x,t) which satisfies (0-2), (0-3), (0-4) and a 2 u < at 2 , w -u > +a( u, w -u) ;:::: < J, w -u > (0-6) for every wE G, for all t E (0, T).