Fractional Sobolev regularity for fully nonlinear elliptic equations

TL;DR

This paper establishes higher-order fractional Sobolev regularity for solutions of fully nonlinear elliptic equations with unbounded source terms, revealing a fractional diffusion feature in such processes.

Contribution

It proves the existence of a universal fractional regularity exponent for viscosity solutions of fully nonlinear elliptic equations with L^p sources.

Findings

Solutions belong to W^{1+ε,p} for some universal ε

Fractional Laplacian of solutions is in L^p

Regularity results depend only on ellipticity and dimension

Abstract

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number $0< \varepsilon <1$, depending only on ellipticity constants and dimension, such that if $u$ is a viscosity solution of $F(D^2u) = f(x) \in L^p$, then $u\in W^{1+\varepsilon,p}$, with appropriate estimates. Our strategy suggests a sort of fractional feature of fully nonlinear diffusion processes, as what we actually show is that $F(D^2u) \in L^p \implies (-\Delta)^\theta u \in L^p$, for a universal constant $\frac{1}{2} < \theta <1$. We believe our techniques are flexible and can be adapted to various models and contexts.