tailieunhanh - Báo cáo hóa học: " The existence of solutions to the nonhomogeneous A-harmonic equation"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : The existence of solutions to the nonhomogeneous A-harmonic equation | Li et al. Journal of Inequalities and Applications 2011 2011 80 http content 2011 1 80 RESEARCH Journal of Inequalities and Applications a SpringerOpen Journal Open Access The existence of solutions to the nonhomogeneous - -harmonic equation Guanfeng Li Yong Wang and Gejun Bao Correspondence mathwy@hit. Department of Mathematics Harbin Institute of Technology Harbin 150001 People s Republic of China Abstract In this paper we introduce the obstacle problem about the nonhomogeneous -harmonic equation. Then we prove the existence and uniqueness of solutions to the nonhomogeneous -harmonic equation and the obstacle problem. Keywords the obstacle problem the nonhomogeneous A-harmonic equation existence and uniqueness of solutions 1 Introduction In this paper we study the nonhomogeneous -harmonic equation div x V u x f x where R X R R is an operator and f is a function satisfying some assumptions given in the next section. We give the definition of solutions to the nonhomoge-neous -harmonic equation and the obstacle problem. In the mean time we show some properties of their solutions. Then we prove the existence and uniqueness of solutions to the Dirichlet problem for the nonhomogeneous -harmonic equation with Sobolev boundary values. Let R be the real Euclidean space with the dimension n. Throughout this paper all the topological notions are taken with respect to Rn. E F means that E is a compact subset of F. C O is the set of all continuous functions u o R. sptu is the smallest closed set such that u vanishes outside sptu. Ck p ỉì R the kth - derivative of p is continuous C0 p e Ck a sptp g C n d a k i and C 0 Q p e C Q sptp Ễ Q . Let Lp O o R Im. p dx and Lp O Rn o Rn Im. p dx 1 p . Denote the norm of Lp o and Lp O Rn by p Springer 2011 Li et al licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http licenses by which .