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Báo cáo hóa học: " Research Article Local Regularity Results for Minima of Anisotropic Functionals and Solutions of Anisotropic Equations"

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 Local Regularity Results for Minima of Anisotropic Functionals and Solutions of Anisotropic Equations | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 835736 11 pages doi 10.1155 2008 835736 Research Article Local Regularity Results for Minima of Anisotropic Functionals and Solutions of Anisotropic Equations Gao Hongya 1 2 Qiao Jinjing 1 Wang Yong 3 and Chu Yuming4 1 College of Mathematics and Computer Science Hebei University Baoding 071002 China 2 Study Center of Mathematics of Hebei Province Shijiazhuang 050016 China 3 Department of Mathematics Chengde Teachers College for Nationalities Chengde 067000 China 4Faculty of Science Huzhou Teachers College Huzhou 313000 China Correspondence should be addressed to Gao Hongya hongya-gao@sohu.com Received 13 July 2007 Accepted 21 November 2007 Recommended by Alberto Cabada This paper gives some local regularity results for minima of anisotropic functionals I u Q ỊQf x u Dù dx u e W 1 - Q and for solutions of anisotropic equations -divA x u Du -y N 1 df loc i 1 ồXi u e W q Q which can be regarded as generalizations of the classical results. Copyright 2008 Gao Hongya et al. 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 be an open bounded subset of RN N 2. Let qi 1 i 1 . N. Denote q max qi p min qi q 1 -1 y . i fN r 1 1 N q N t- qi Throughout this paper we will make use of the anisotropic Sobolev space W.1 Q iv e Lq Q e L Q Vi 1 . nL loc loc dx- loc 1.1 1.2 Let x0 e Q and t 0 we denote by Bt the ball of radius t centered at x0. For functions u and k 0 let Ak x e Q u x k Ak t Ak n Bt. Moreover if p 1 then p is always the real number p p - 1 and if s N s is always the real number satisfying 1 s 1 s - 1 N. This paper mainly considers the functions u minimizing the anisotropic functionals Iu Q f x u Du dx u e W 1 Q loc Q 1.3 2 Journal of Inequalities and Applications and weak solutions of the .