Optimal $C^{1,\alpha}$ regularity up to the boundary for fully nonlinear elliptic equations with double phase degeneracy

TL;DR

This paper proves optimal boundary regularity for solutions of fully nonlinear elliptic equations with double phase degeneracy, using boundary estimates and stability methods.

Contribution

It introduces a new approach to establish boundary $C^{1,eta}$ regularity for degenerate elliptic equations with oblique boundary conditions.

Findings

Established optimal $C^{1,eta}$ regularity up to the boundary.

Determined the optimal Hölder exponent for quasiconvex/quasiconcave operators.

Improved regularity results at vanishing source points.

Abstract

In this paper we establish optimal $C^{1,\alpha}$ regularity up to the boundary for viscosity solutions of fully nonlinear elliptic equations with double phase degeneracy law and oblique boundary conditions. The approach developed here relies on first deriving uniform boundary H\"older estimates for perturbed models with oblique boundary data in ``almost $C^{1}$-flat'' domains. Building upon these estimates, the desired regularity is obtained through a compactness and stability framework for viscosity solutions. As a byproduct of our analysis, we determine the optimal H\"older exponent for solutions when the governing operator is quasiconvex or quasiconcave. In addition, we establish an improved regularity result along vanishing points of the source term.