Asymptotics and regularity of flat solutions to fully nonlinear elliptic problems

TL;DR

This paper proves regularity estimates for flat solutions to fully nonlinear elliptic equations, showing they are locally $C^{2,eta}$ when coefficients are $C^{0,eta}$, and $C^{1, ext{Log-Lip}}$ with continuous data.

Contribution

It establishes new regularity results for flat solutions of non-convex fully nonlinear elliptic equations under minimal regularity assumptions.

Findings

Flat solutions are locally $C^{2,eta}$ if coefficients and source are $C^{0,eta}$.

Flat solutions are locally $C^{1, ext{Log-Lip}}$ with merely continuous data.

Provides regularity estimates for non-convex fully nonlinear elliptic equations.

Abstract

In this work we establish local $C^{2,\alpha}$ regularity estimates for flat solutions to non-convex fully nonlinear elliptic equations provided the coefficients and the source function are of class $C^{0,\alpha}$. For problems with merely continuous data, we prove that flat solutions are locally $C^{1,\text{Log-Lip}}$.