Weighted analytic regularity for the integral fractional Laplacian in polygons

TL;DR

This paper establishes weighted analytic regularity results for solutions to the Dirichlet problem involving the integral fractional Laplacian in polygonal domains, using extension techniques and local analysis.

Contribution

It introduces a novel approach combining the Caffarelli-Silvestre extension with localized regularity estimates to analyze fractional Laplacian problems in polygons.

Findings

Weighted analytic regularity of solutions proved

Localization techniques effectively handle polygonal domains

Regularity results applicable to problems with analytic data

Abstract

We prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian in polygons with analytic right-hand side. We localize the problem through the Caffarelli-Silvestre extension and study the tangential differentiability of the extended solutions, followed by bootstrapping based on Caccioppoli inequalities on dyadic decompositions of vertex, edge, and edge-vertex neighborhoods.