Sharp hessian integrability estimates for nonlinear elliptic equations: an asymptotic approach

TL;DR

This paper proves sharp second-order regularity estimates for solutions of fully nonlinear elliptic equations using an asymptotic approach, extending regularity results under minimal assumptions on the operator.

Contribution

It introduces a novel asymptotic method to establish $W^{2,p}$ regularity for viscosity solutions based on the convexity of the operator's recession function.

Findings

Established sharp $W^{2,p}$ estimates for viscosity solutions.

Extended regularity results to operators with variable coefficients.

Proved the density of $W^{2,p}$ solutions among all continuous viscosity solutions.

Abstract

We establish sharp $W^{2,p}$ regularity estimates for viscosity solutions of fully nonlinear elliptic equations under minimal, asymptotic assumptions on the governing operator $F$. By means of geometric tangential methods, we show that if the {\it recession} of the operator $F$ -- formally given by $F^*(M):=\infty^{-1} F(\infty M)$ -- is convex, then any viscosity solution to the original equation $F(D^2u) = f(x)$ is locally of class $W^{2,p}$, provided $f\in L^p$, $p>d$, with appropriate universal estimates. Our result extends to operators with variable coefficients and in this setting they are new even under convexity of the frozen coefficient operator, $M\mapsto F(x_0, M)$, as oscillation is measured only at the recession level. The methods further yield BMO regularity of the hessian, provided the source lies in that space. As a final application, we establish the density of $W^{2,p}$ solutions within the class of all continuous viscosity solutions, for generic fully nonlinear operators $F$. This result gives an alternative tool for treating common issues often faced in the theory of viscosity solutions.