Potential estimates for fully nonlinear elliptic equations with bounded ingredients

TL;DR

This paper investigates the regularity of solutions to fully nonlinear elliptic equations with bounded measurable ingredients, establishing conditions for local Lipschitz continuity and gradient continuity using nonlinear potential estimates.

Contribution

It introduces new gradient-regularity estimates for $L^p$-viscosity solutions based on nonlinear potential theory, extending existing regularity results.

Findings

Conditions for local Lipschitz continuity of solutions

Continuity of the gradient under certain conditions

Connections to recent advances in nonlinear potential estimates

Abstract

We examine $L^p$-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $p_0<p<d$, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We briefly survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from -- and are inspired by -- fundamental facts in the theory of $L^p$-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi and Giuseppe Mingione [10].