tailieunhanh - Báo cáo hóa học: "Research Article Discontinuous Variational-Hemivariational Inequalities Involving the p-Laplacian"

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 Discontinuous Variational-Hemivariational Inequalities Involving the p-Laplacian | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 13579 11 pages doi 2007 13579 Research Article Discontinuous Variational-Hemivariational Inequalities Involving the p-Laplacian Patrick Winkert Received 6 August 2007 Accepted 25 November 2007 Recommended by M. Garcia-Huidobro We deal with discontinuous quasilinear elliptic variational-hemivariational inequalities. By using the method of sub- and supersolutions and based on the results of S. Carl we extend the theory for discontinuous problems. The proof of the existence of extremal solutions within a given order interval of sub- and supersolutions is the main goal of this paper. In the last part we give an example of the construction of sub- and supersolutions. Copyright 2007 Patrick 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 do. As V W1 p Q and V0 W1 p Q 1 p 00 we denote the usual Sobolev spaces with their dual spaces V W1 p Q and V0 W-1 q Q respectively q is the Holder conjugate of p . In this paper we consider the following elliptic variational-hemivariational inequality u e K - Apu F u v - u J j0 u v - u dx 0 Vv e K where j0 s r denotes the generalized directional derivative of the locally Lipschitz function j R R at s in the direction r given by j0 s r limsup y -s M 0 j y tr - j y t cf. 1 Chapter 2 and K c V0 is some closed and convex subset. The operator ApU div Vu p-2Vu is the p-Laplacian 1 p o and F denotes the Nemytskij operator 2 Journal of Inequalities and Applications related to the function f Q X R X R R given by F u x f x u x u x . In 2 the method of sub- and supersolutions was developed for variational-hemivaria-tional inequalities of the form with F u f e V0 . The aim of this paper is