${{W}^{2,\delta}}$ Estimates for Solution Sets of Fully Nonlinear Elliptic inequalities on ${C^{1,\alpha}}$ Domains

TL;DR

This paper develops boundary $W^{2, ext{delta}}$ estimates for solutions of fully nonlinear elliptic inequalities on $C^{1, ext{alpha}}$ domains, using interior estimates and Whitney decomposition without boundary straightening.

Contribution

It introduces a novel approach to boundary regularity by deriving boundary estimates from interior ones, avoiding boundary straightening techniques.

Findings

Established boundary $W^{2, ext{delta}}$ estimates for solutions on $C^{1, ext{alpha}}$ domains.

Extended interior $W^{2, ext{delta}_0}$ estimates to boundary estimates via Whitney decomposition.

Provided regularity results for solutions with $L^p$ data on non-smooth domains.

Abstract

In this paper, we establish boundary $W^{2,\delta}$ estimates for $u\in S(\lambda,\Lambda,f)$ on $C^{1,\alpha}$ domains with $f\in L^p$ as $n<p<\infty$ and $C^{1,\alpha}$ boundary values. Instead of straightening out the boundary, our main idea is to obtain boundary $W^{2,\delta}$ estimates from interior $W^{2,\delta_0}$ estimates and Whitney decomposition for some $\delta\leq \delta_0$.