Boundary Regularity for Fully Nonlinear Singular Elliptic Equations

Abstract

We study boundary regularity for the fully nonlinear singular elliptic equation of the form $|Du|^\gamma F(D^2u)=f$, where $-1<\gamma<0$. First, we prove pointwise boundary $C^{1,\alpha}$ estimates with pointwise $C^{1,\alpha}$ Dirichlet data, which weakens the $C^2$ domain smoothness requirement from Birindelli and Demengel~\cite{BD10}. The proof uses compactness and perturbation, without flattening the boundary. Second, we extend the $W^{2,\delta}$ estimates of Li and Li~\cite{LiLi17} from balls to $C^{1,\alpha}$ domains with $C^{1,\alpha}$ Dirichlet data{, for $0<\delta\le\delta_0<1$. Here $\delta_0$ is an exponent in the interior $W^{2,\delta_0}$ estimate}. The proof combines pointwise boundary $C^{1,\alpha}$ estimates, interior $W^{2,\delta_0}$ estimates, and a Whitney decomposition.