Properties Of Very Weak Solutions For Nonhomogeneous A-harmonic Equations

The A-harmonic equation can describe the complex phenomenon in the field of electromagnetic,the theory of relativity,theory of elasticity and nonlinear potential theory more accurately than other equations,which makes A-harmonic equation has been widely applied in these areas and its' related fields.Furthermore,there exists a close relationship between quasi regular mapping and A-harmonic equation,the study results of A-harmonic equation can provide the theoretical foundation and research tools for the establishing of quasi regular mapping's theory.And quasi regular mapping as a hot research subject has always been keen by many mathematical researchers.Thus,the theoretical research of A-harmonic equation attracts many mathematical researchers' eyes.This paper mainly considers the property of very weak solutions to inhomogeneous Aharmonic equation(1)Firstly,by analyzing the problem,yielding the Hodge decomposition on very weak solution,complying the property of Hardy-Littlewood's maximum function with Young inequality,Holder inequality,etc,we establish the comparison principle of the very weak solution to inhomogeneous A-harmonic equation,and obtain the following conclusion.Theorem 1(comparison principle)Suppose that is bounded domain,there exists a constant,when,such that in the sense of Sobolev,the two very weak solution to inhomogeneous quasilinear A-harmonic equation(1)under the structural conditions(H1)-(H3)satisfy: if on the boundaries of the domain,there has(or),then in the domain,the following inequality(or)holds almost everywhere.Then,applying Hodge decomposition,Sobolev embedding theorem and the research technique in the study of regularity theory,we discuss the regularity theory of very weak solution to inhomogeneous quasilinear A-harmonic equation(1)under certain structural conditions.That isTheorem 2(regular theorem)Suppose that,there exists an integral index,such that every very weak solution to inhomogeneous quasilinear A-harmonic equation(1)under the structural conditions(H1)-(H3)belongs to,which means that is a classical weak solution.