A Study For Weak Solutions Of Anisotropic Elliptic Equation

Elliptic equations play a very important role in the theoretical research of partial differential equations, which show a wide application background in mathematics (dif-ferential geometry, quasiregular mapping etc), engineering physics (fluid mechanics etc) and biology (population stability etc) etc. The theory of elliptic equations mainly includes uniqueness, stability and regularity. This paper mainly discusses the local regularity for weak solutions of anisotropic elliptic equations, and the local regularity and stability of weak solution of obstacle problem to anisotropic elliptic equations.The first chapter introduces the research background of the elliptic equations and its research status all over the world. From isotropic elliptic equation to anisotropic elliptic equation, from the classical solutions of equation to weak solutions further to very weak solutions, people gradually investigate the property of this kind of solutions of elliptic equations. In view of the weak solutions of anisotropic elliptic equation, we put forward some problems need to be solved and the major research methods as well as some prob-lems are to be studied.The second chapter mainly studies the local regularity of weak solutions of Kψ,θ(pi)-obstacle problems for a class of nonlinear elliptic equations-divA(x, Du)=B(x, u, Du). In this chapter, we obtain the local regularity for weak solutions of obstacle problems of anisotropic elliptic equation, through constructing appropriate anisotropic test function, and using the Sobolev inequality, Holder’s inequality and Young’s inequality.The third chapter is divided into two parts. One part we consider the anisotropic A-harmonic equation, by using the anisotropic embedding theorem and some basic in-equalities, the uniqueness and the stability of weak solution of the obstacle problem are obtained. Another part is to study the minimizer of weak solution of obstacle problem for the anisotropic A-harmonic equation, which is a foundation to study the weak solution of anisotropic elliptic equation.The fourth chapter study the local regularity of the weak solutions of A-harmonic equation. By constructing an anisotropic test function, using the anisotropic embedding theorem and related lemmas, the result of regularity of weak solution is obtained in this chapter.