Regularity And Uniqueness Of Very Weak Solution For Degenerate Elliptic Equations

The A— harmonic equations are an important extension of p—harmonic equation in Rⁿ for p > 1. They are intimately connected to the fields such as potential theory, quasi-regular mapping(quasi-conformal mappings) and the theory of elasticity. Many interesting results concerning geometric and analytic properties of solutions to the A— harmonic type equation have been established recently. Since 1990s, with the begin of the concept of very weak solution proposed, the relating research about them have been widespread emerged. This paper mainly study the regularity and uniqueness results of very weak solution for several classes of degenerate elliptic equations, so-called A—harmonic type equations, which is divided into four chapters :In chapter one, we describe the background of the very weak solution, the preliminary inequalities and the relation between "Caccippoli inequality" and very weak solutions in the modern theory of partial differential equations.In chapter two, we discuss the uniqueness of very weak solutions for the following A-harmonic type equation,-divA(x,(?)u) = f(x),where the operator A(x,·) satisfies the following three assumptions: (H1) the degenerate ellipticity〈A(x,ξ₁) - A(x,ξ₂),ξ₁ - ξ₂〉 ≥ β(|ξ₁| + |ξ₂|)^p-2|ξ₁ - ξ₂|²;(H2) the Lipschitz type inequality|A(x,ξ₁) - A(x,ξ₂)| ≤σ|ξ₁- ξ₂|(|ξ₁| + |ξ₂|)^p-2;(H3) the homogeneity conditionA(x,λξ) = |λ|^p-2λA(x,ξ).We don't use usual harmonic analysis method which in literature[3],but adopt Hodge decomposition and estimates which used to study quasi-mapping by Iwaniec. T. We obtain the conclusion ,promote and refine the similiar problem which in literature [1]...