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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: MAXIMUM PRINCIPLES FOR A CLASS OF NONLINEAR SECOND-ORDER ELLIPTIC BOUNDARY VALUE PROBLEMS IN DIVERGENCE FORM CRISTIAN ENACHE | MAXIMUM PRINCIPLES FOR A CLASS OF NONLINEAR SECOND-ORDER ELLIPTIC BOUNDARY VALUE PROBLEMS IN DIVERGENCE FORM CRISTIAN ENACHE Received 22 January 2006 Accepted 26 March 2006 For a class of nonlinear elliptic boundary value problems in divergence form we construct some general elliptic inequalities for appropriate combinations of u x and Vu 2 where u x are the solutions of our problems. From these inequalities we derive using Hopf s maximum principles some maximum principles for the appropriate combinations of u x and Vu 2 and we list a few examples of problems to which these maximum principles may be applied. Copyright 2006 Cristian Enache. 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 u x be the classical solution of the following nonlinear boundary value problems g u Vu 2 u i j h x f u Vu 2 0 x e Q 1.1 u 0 x e dQ 1.2 where Q is a bounded domain in RN N 2 with smooth boundary dQ e C2 and f g and h are given functions assumed to satisfy the following conditions f h 0 g 0 f h e C1 g e C2. 1.3 Moreover we assume that 1.1 is uniformly elliptic that is we impose throughout the strong ellipticity condition G u s g u s 2s zg 0 s 0 x e Q. Os 1.4 Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID 64543 Pages 1-13 DOI 10.1155 BVP 2006 64543 2 Maximum principles for a class of elliptic problems Under these assumptions a minimum principle for the solutions u x of the nonlinear equation 1.1 follows immediately that is u x must assume its minimum value on du. Sufficient conditions on the data for the existence of classical solutions of the nonlinear equation 1.1 are known and have been well studied in the literature. See for instance LadyZenskaja and Ural ceva 5 for an account on this topic. Consequently we will tacitly assume the existence of classical solutions of the
