The Very Weak Solutions Of Anisotropic Obstacle Problems And Their Properties

A-harmonic equation is a kind of important and typical elliptic equation. It has deep physical and mechanical background, and has application value in many other disciplines. The research for the properties of A-harmonic equation weak solutions and very weak solutions has gained a lot of conclusions and a certain research has been gained for the weak solutions of anisotropic obstacle problems. This paper is to study the very weak solutions and those properties of the anisotropic obstacle problems.On the basis of weak solutions of the anisotropic obstacle problems and the very weak solutions of the obstacle problems, this paper mainly gives the definitions of the very weak solutions of A-harmonic equations anisotropic obstacle problems. Under certain conditions, the local boundedness, local regularity and uniqueness of the very weak solutions of the anisotropic obstacle problems to the A-harmonic equations are obtained by constructing special test functions and using the Holder’s inequality, the Young inequality, Poincare inequality, reverse Holder’s inequality, Fatou lemma and the theory analysis in Sobolev space.The properties are as follows. The local boundedness of very weak solutions of the anisotropic obstacle problems to the homogeneous A-harmonic equations-divA(x,Vu)=0. The local regularity of very weak solutions of the anisotropic obstacle problems to the non homogenous A-harmonic equations-divA(x,Vu)=-divf. The local boundedness of very weak solutions of the anisotropic obstacle problems to the A-harmonic equations-divA(x,u,(?)w)=-divf. The local regularity of very weak solutions of the anisotropic obstacle problems to the A-harmonic equations-divA(x,(?)u)=0. The uniqueness of very weak solutions of the anisotropic obstacle problems to the A-harmonic equations-divA(x,(?)u)=0.The above researches enrich the theory basis of A-harmonic equations.