Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article General Comparison Principle for Variational-Hemivariational Inequalities | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 184348 29 pages doi 2009 184348 Research Article General Comparison Principle for Variational-Hemivariational Inequalities Siegfried Carl and Patrick Winkert Department of Mathematics Martin-Luther-University Halle-Wittenberg 06099 Halle Germany Correspondence should be addressed to Patrick Winkert patrick@ Received 13 March 2009 Accepted 18 June 2009 Recommended by Vy Khoi Le We study quasilinear elliptic variational-hemivariational inequalities involving general Leray-Lions operators. The novelty of this paper is to provide existence and comparison results whereby only a local growth condition on Clarke s generalized gradient is required. Based on these results in the second part the theory is extended to discontinuous variational-hemivariational inequalities. Copyright 2009 S. Carl and P. Winkert. 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 Let Q c RN N 1 be a bounded domain with Lipschitz boundary ÔQ. By W1 p Q and W01 P Q 1 p TO we denote the usual Sobolev spaces with their dual spaces W1 p Q and W 1 q Q respectively where q is the Holder conjugate satisfying 1 p 1 q 1. We consider the following elliptic variational-hemivariational inequality. Find u K such that Au F u v - u J j0 - u v - ù dx J jVyu ỴV - ỵu dơ 0 Vv e K where j0 x s r k 1 2 denotes the generalized directional derivative of the locally Lipschitz functions s jk x s at s in the direction r given by t i . limsupId t r - WW k -- 1 2 y sztịQ 2 Journal of Inequalities and Applications cf. 1 Chapter 2 . We denote by K a closed convex subset of W 1 p Q and A is a second-order quasilinear differential operator in divergence form of Leray-Lions type given by Au x - r ai x u x Vu xY . i 1 dxi .
