Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

TL;DR

This paper advances the theory of symmetrization for fractional Laplacian operators, providing optimal estimates and inequalities for elliptic and parabolic equations, with applications to the fractional Faber-Krahn inequality.

Contribution

It extends symmetrization theory to restricted fractional Laplacians on bounded domains and offers new proofs for the fractional Faber-Krahn inequality.

Findings

Established optimal concentration comparison inequalities for fractional Laplacian equations.

Extended symmetrization theory to bounded domains with zero Dirichlet conditions.

Provided new proofs of the fractional Faber-Krahn inequality.

Abstract

We develop further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. The theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. In this paper we extend the theory for the so-called \emph{restricted} fractional Laplacian defined on a bounded domain $\Omega$ of $\mathbb R^N$ with zero Dirichlet conditions outside of $\Omega$. As an application, we derive an original proof of the corresponding fractional Faber-Krahn inequality. We also provide a more classical variational proof of the inequality.