Boundary regularity for viscosity solutions of fully nonlinear elliptic equations

TL;DR

This paper establishes boundary regularity results for viscosity solutions of fully nonlinear elliptic equations, extending classical estimates to less regular solutions and providing detailed boundary behavior descriptions.

Contribution

It proves boundary regularity and asymptotic expansion results for viscosity solutions of nonconvex fully nonlinear elliptic equations, extending Krylov estimates.

Findings

Viscosity solutions are $C^{1,eta}$ on the boundary for inequalities with Pucci operators.

Solutions to Dirichlet problems are $C^{2,eta}$ on the boundary.

Asymptotic expansions at the boundary are established.

Abstract

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is $C^{1,\alpha}$ on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is $C^{2,\alpha}$ on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.