Global fractional Sobolev regularity for fully nonlinear elliptic equations

TL;DR

This paper establishes global fractional Sobolev regularity for solutions to fully nonlinear elliptic equations, showing solutions are more regular than previously known without extra operator assumptions.

Contribution

It proves that viscosity solutions to fully nonlinear elliptic equations are globally in fractional Sobolev spaces with order greater than one, under minimal assumptions.

Findings

Solutions satisfy fractional Laplacian equations

Solutions are in $W^{oldsymbol{ ext{ extit{ extgamma}}, p}}$ for $ ext{ extgamma} extgreater 1$

Regularity holds without convexity or concavity assumptions

Abstract

We investigate fractional regularity estimates up to the boundary for solutions to fully nonlinear elliptic equations with measurable ingredients. Specifically, under the assumption of uniform ellipticity of the operator, we demonstrate that viscosity solutions to a second-order operator satisfy a fractional Laplacian equation. This result implies that the solutions are globally of class $W^{\gamma, p}$, for $\gamma \in (1,2)$, with appropriate estimates. Consequently, these solutions exhibit differentiability of order strictly greater than one, without requiring any additional assumptions regarding the operator, such as convexity or concavity.