A Probabilistic proof of the breakdown of Besov regularity in $L$-shaped domains

TL;DR

This paper uses probabilistic methods to analyze the loss of Besov regularity in solutions to Poisson problems in L-shaped domains, providing integral representations and confirming classical angle-dependent regularity breakdowns.

Contribution

It introduces a probabilistic approach to study solution regularity in L-shaped domains, offering new integral representations and confirming classical regularity results.

Findings

Probabilistic integral representations for solutions

Confirmation of angle-dependent Besov regularity breakdown

New insights into solution smoothness in non-smooth domains

Abstract

{We provide a probabilistic approach in order to investigate the smoothness of the solution to the Poisson and Dirichlet problems in $L$-shaped domains. In particular, we obtain (probabilistic) integral representations for the solution. We also recover Grisvard's classic result on the angle-dependent breakdown of the regularity of the solution measured in a Besov scale.