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Báo cáo hóa học: "BOUNDARY REGULARITY OF WEAK SOLUTIONS TO NONLINEAR ELLIPTIC OBSTACLE PROBLEMS"

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: BOUNDARY REGULARITY OF WEAK SOLUTIONS TO NONLINEAR ELLIPTIC OBSTACLE PROBLEMS | BOUNDARY REGULARITY OF WEAK SOLUTIONS TO NONLINEAR ELLIPTIC OBSTACLE PROBLEMS MENG JUNXIA AND CHU YUMING Received 25 April 2005 Revised 10 September 2005 Accepted 14 September 2005 We study the boundary regularity of weak solutions to nonlinear obstacle problem with C1 -obstacle function and obtain the Cloc1 a boundary regularity. Copyright 2006 M. Junxia and C. Yuming. 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 We consider the following variational inequality u e J A x Vu Vv -Vu dx J H x u Vu v - u dx J F x u Vv -Vu dx for all v e v e w0zp D v y a.e. in D . Here D is a bounded domain in RN N 2 with Lipschitz boundary 2 p N. A x ỉ D X RN RN satisfies the following conditions i A is a vector valued function the mapping x A x ỉ is measurable for all ỉ e RN ỉ A x ỉ is continuous for a.e. x e D ii the homogeneity condition A x tỉ t t p-2A x ỉ t e R t 0 iii the monotone inequality A x ỉ - A x z ỉ - z a ỉ - z p iv h aij dAi x h dxj T1 h p-1 v SNj 1 aiiỉiỉj T2 h p-2 ỉ 2 vi A x ỉ - A y ỉ b1 1 ỉ p-1 x-y a0 vii A x ỉ - A x n b2 ỉ p-2ỉ - n p-2n where aij dAi dhj a b1 b2 T1 T2 are positive constants. Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID 72012 Pages 1-15 DOI 10.1155 BVP 2006 72012 2 Boundary regularity We assume that H x u A F x u Fị x u 1 j N in 1.1 are of the form H x u Vu c Vu p r u r 1 g x 1.2 F x u c u q p h x 1.3 where p q r r r r - 1 p p p - 1 and if 2 p N r Np N - p while if p N then r can be some sufficiently large positive number. Higher regularity of the weak solution to the p-Laplacian obstacle problem I u infị J Vu pdx u e K y 1.4 where K y v e w1 p Q v y a.e. 1.5 has been studied by various authors. In the case when y is assumed to have only minimal regularity properties it was shown by 8 11 that the solution of 1.1 is continuous. In .