${{W}^{2,\delta}}$ Estimates for Solution Sets of Fully Nonlinear Elliptic inequalities on ${C^{1,\alpha}}$ 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 $\delta\leq \delta_0$.