Boundary Regularity for Fully Nonlinear Degenerate Elliptic Equations

Abstract

This paper investigates fully nonlinear degenerate elliptic equations of the form $|Du{|}^{\gamma }F(D^{2}u)=f$ with $\gamma >0$ {and $F(0)=0$}. Using a unified compactness and perturbation framework, we {establish} $C^1$ regularity on $C^{1,\mathrm{Dini}}$ domains and {$C^{1,{\alpha }_0}$ regularity on $C^{1,\alpha }$ domains}, {where ${\alpha }_0=\min \{\alpha ,1/(1+\gamma )\}$}. {In addition}, we construct a counterexample to show that {$W^{2,\delta }$ regularity does not hold for any $\delta >0$}.

arXiv comment18 pages