Journal of Inequalities and Applications 
Volume 7 (2002), Issue 4, Pages 479-502
doi:10.1155/S1025583402000243

Extremal solutions of a class of dynamic boundary hemivariational inequalities

Siegfried Carl1 and R. P. Gilbert2

 1Fachbereich Mathematik und Informatik, Institut für Analysis, Martin-Luther-Universität Halle-Wittenberg, Halle D-06099, Germany
2Department of Mathematical Sciences, University of Delaware, Newark 19716, Delaware, USA
Received 10 June 2000; Revised 12 November 2000

Abstract

In this paper we consider a semilinear initial boundary value problem in a bounded cylindrical domain under flux conditions described by Clarke’s generalized gradient of some locally Lipschitz function j:ℝ→ℝ. Our main goal is to prove the existence of extremal solutions within a sector formed by a pair of appropriately defined upper and lower solutions when the function j is of d.c. type, which means that j can be represented as the difference of convex functions jk:ℝ→ℝ, k=1,2. The main tools used in the proofs are results on nonlinear evolution equations, compact embeddings, comparison, truncation and regularization techniques.