Universal moduli of continuity for solutions to fully nonlinear elliptic equations

TL;DR

This paper establishes universal optimal moduli of continuity for solutions to fully nonlinear elliptic equations, linking regularity to the integrability properties of the source term and providing sharp estimates across various function spaces.

Contribution

It introduces new sharp regularity estimates for viscosity solutions based on minimal integrability assumptions of the source function, including critical and borderline cases.

Findings

Sharp Log-Lipschitz estimate based on $L^n$ norm of $f$

Optimal $C^{1, ext{Log-Lip}}$ regularity for $f$ in BMO

Interior estimates for $f$ in $L^{n- ext{epsilon}}$ with optimal exponent

Abstract

This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations $F(X, D^2u) = f(X)$, based on weakest integrability properties of $f$ in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on $u$ based on the $L^n$ norm of $f$, which corresponds to optimal regularity bounds for the critical threshold case. Optimal $C^{1,\alpha}$ regularity estimates are delivered when $f\in L^{n+\epsilon}$. The limiting upper borderline case, $f\in L^\infty$, also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under convexity assumption on $F$, that $u \in C^{1,\mathrm{Log-Lip}}$, provided $f$ has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior \textit{a priori} estimates on the $C^{0,\frac{n-2\epsilon}{n-\epsilon}}$ norm of $u$ based on the $L^{n-\epsilon}$ norm of $f$, where $\epsilon$ is the Escauriaza universal constant. The exponent $\frac{n-2\epsilon}{n-\epsilon}$ is optimal. When the source function $f$ lies in $L^q$, $n > q > n-\epsilon$, we also obtain the exact, improved sharp H\"older exponent of continuity.