${W}^{2,p}$ Estimates for Elliptic Equations on $C^{1,\alpha}$ Domains

TL;DR

This paper introduces a novel method to derive boundary W^{2,p} estimates for elliptic equations on C^{1,α} domains by leveraging interior estimates through Whitney decomposition, applicable to both linear and nonlinear cases.

Contribution

The paper presents a new approach to obtain boundary W^{2,p} estimates from interior estimates for elliptic equations on C^{1,α} domains, extending to fully nonlinear equations.

Findings

Established boundary W^{2,p} estimates for linear elliptic equations in C^{1,α} domains.

Extended the method to fully nonlinear elliptic equations.

Demonstrated the effectiveness of Whitney decomposition in boundary estimate derivation.

Abstract

In this paper, a new method is represented to investigate boundary $W^{2,p}$ estimates for elliptic equations, which is, roughly speaking, to derive boundary $W^{2,p}$ estimates from interior $W^{2,p}$ estimates by Whitney decomposition. Using it, $W^{2,p}$ estimates on $C^{1,\alpha}$ domains are obtained for nondivergence form linear elliptic equations and further more, fully nonlinear elliptic equations are also considered.