Symmetrization for fractional Neumann problems

TL;DR

This paper advances the understanding of nonlocal PDEs with Neumann boundary conditions by applying symmetrization techniques to derive new estimates and regularity results for fractional elliptic and parabolic equations.

Contribution

It introduces novel symmetrization-based estimates for fractional Neumann problems using the Stinga-Torrea extension approach, enhancing regularity and concentration analysis.

Findings

New mass concentration comparison estimates for fractional PDE solutions.

Sharp regularity estimates for elliptic fractional Neumann problems.

A parabolic symmetrization result derived from elliptic concentration estimates.

Abstract

In this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and parabolic PDEs with Neumann boundary conditions. These results are achieved by employing suitable symmetrization arguments to the Stinga-Torrea local extension problems, corresponding to the fractional boundary value problems considered. Sharp estimates are obtained first for elliptic equations and a certain number of consequences in terms of regularity estimates is then established. Finally, a parabolic symmetrization result is covered as an application of the elliptic concentration estimates in the implicit time discretization scheme.