Abstract
Abstract Let w be a Muckenhoupt A 2(ℝ n ) weight and Ω a bounded Reifenberg flat domain in ℝ n . Assume that p (·):Ω → (1, ∞) is a variable exponent satisfying the log-Hölder continuous condition. In this article, the authors investigate the weighted W 1, p (·)(Ω, w)-regularity of the weak solutions of second order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under the assumption that the degenerate coefficients belong to weighted BMO spaces with small norms.
Highlights
The authors investigate the weighted W, p(·)(Ω, w)-regularity of the weak solutions of second order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under the assumption that the degenerate coe cients belong to weighted BMO spaces with small norms
We establish the weighted interior and boundary gradient approximation estimates of the weak solution u of degenerate elliptic equations, which play a key role in the proof of Theorem 1.7
By the assumption that v and u solve, respectively, (4.2) and (4.1), we conclude that u − v is a weak solution of div [A∇(u − v)] = div [F − (A − A )∇v] in B, where u = u − u B, w
Summary
We consider the following second order degenerate elliptic equation in divergence form with. A : Rn → Rn×n is a symmetric matrix of measurable functions {aij}ni, j= on Rn satisfying the degenerate elliptic condition, namely, there exists a positive constant Λ ∈ [ , ∞) such that, for any ξ ∈ Rn and almost every x ∈ Rn, Λ− w(x)|ξ | ≤ A(x)ξ , ξ ≤ Λw(x)|ξ |. We always let Λ be as in (1.2) and A a symmetric matrix. Recall that a non-negative locally integrable function w on Rn is said to belong to the Muckenhoupt class Ap(Rn), denoted by w ∈ Ap(Rn), if, when p ∈ ( , ∞), p−
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