Partial regularity of solutions of fully nonlinear uniformly elliptic equations

TL;DR

This paper establishes partial regularity results for viscosity solutions of fully nonlinear uniformly elliptic equations, showing they are $C^{2,eta}$ outside a small Hausdorff dimension set, under certain smoothness conditions.

Contribution

It proves that solutions are $C^{2,eta}$ except on a set of small Hausdorff dimension, extending regularity theory for fully nonlinear elliptic equations.

Findings

Solutions are $C^{2,eta}$ outside a set of Hausdorff dimension at most $ ext{dim}- ext{epsilon}$.

The regularity result depends only on the dimension and ellipticity constants.

Combines $W^{2, ext{epsilon}}$ estimates with Savin's quadratic approximation results.

Abstract

We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is $C^{2,\alpha}$ on the compliment of a closed set of Hausdorff dimension at most $\epsilon$ less than the dimension. The equation is assumed to be $C^1$, and the constant $\epsilon > 0$ depends only on the dimension and the ellipticity constants. The argument combines the $W^{2,\epsilon}$ estimates of Lin with a result of Savin on the $C^{2,\alpha}$ regularity of viscosity solutions which are close to quadratic polynomials.