On the existence of smooth solutions for fully nonlinear elliptic equations with measurable "coefficients" without convexity assumptions

TL;DR

This paper demonstrates the existence of smooth solutions for fully nonlinear elliptic equations with measurable coefficients, by constructing an approximating equation that ensures regularity without requiring convexity assumptions.

Contribution

It introduces a method to approximate fully nonlinear elliptic equations with solutions that are continuous and have locally bounded second derivatives, bypassing convexity constraints.

Findings

Existence of smooth solutions for a broad class of nonlinear elliptic equations.

Construction of an approximating equation modifying the original only at large values.

Solutions are continuous with second derivatives locally bounded in smooth domains.

Abstract

We show that for any uniformly elliptic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term one can find an approximating equation which has a unique continuous and having the second derivatives locally bounded solution in a given smooth domain with smooth boundary data. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its derivatives.