Fractional Sobolev regularity for fully nonlinear elliptic equations

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 $0<\epsilon <1$, depending only on ellipticity constants and dimension, such that if $u$ is a viscosity solution of $F(D^{2}u)=f(x)\in L^p$, then $u\in W^{1+\epsilon ,p}$, with appropriate estimates. Our strategy suggests a sort of fractional feature of fully nonlinear diffusion processes, as what we actually show is that $F(D^{2}u)\in L^p⟹(-\Delta {)}^{\theta }u\in L^p$, for a universal constant $\frac{1}{2}<\theta <1$. We believe our techniques are flexible and can be adapted to various models and contexts.