Weighted analytic regularity for the integral fractional Laplacian in polyhedra

TL;DR

This paper establishes weighted analytic regularity of solutions to the fractional Laplacian in polyhedral domains, using extension techniques and localized estimates to handle boundary singularities.

Contribution

It introduces a novel approach combining the Caffarelli-Silvestre extension with boundary neighborhood decompositions to prove weighted analytic regularity.

Findings

Weighted regularity results for solutions in polyhedra

Decomposition of boundary neighborhoods for regularity analysis

Bootstrapping method for higher order derivatives

Abstract

On polytopal domains in $\mathbb{R}^3$, we prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian with analytic right-hand side. Employing the Caffarelli-Silvestre extension allows to localize the problem and to decompose the regularity estimates into results on vertex, edge, face, vertex-edge, vertex-face, edge-face and vertex-edge-face neighborhoods of the boundary. Using tangential differentiability of the extended solutions, a bootstrapping argument based on Caccioppoli inequalities on dyadic decompositions of the neighborhoods provides weighted, analytic control of higher order solution derivatives.