Comparison and regularity results for the fractional Laplacian via symmetrization methods

TL;DR

This paper develops symmetrization techniques to compare solutions of fractional Laplacian boundary problems, leading to sharp estimates and regularity results that extend classical Laplacian theory.

Contribution

It introduces a symmetrization-based comparison method for fractional Laplacian problems, providing new sharp estimates and regularity results.

Findings

Established comparison results for fractional Laplacian solutions

Derived sharp estimates by comparing with radial solutions

Extended classical Laplacian regularity results to fractional case

Abstract

In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well known ones for the classical Laplacian.