Robin harmonic measure with a variable permeability parameter

TL;DR

This paper investigates the Robin boundary value problem in complex domains with variable permeability, establishing the mutual absolute continuity of Robin harmonic measure and surface measure, extending previous results to non-constant Robin parameters.

Contribution

It generalizes prior work by proving mutual absolute continuity of Robin harmonic measure with surface measure for variable permeability in complex domains.

Findings

Mutual absolute continuity of Robin harmonic measure and surface measure.

Extension of results to non-constant Robin parameters.

Analysis in NTA domains with Ahlfors-David regular boundary.

Abstract

In this paper we study the behavior of the solutions to the Robin problem in bounded $1$-sided NTA domains with Ahlfors-David regular boundary, generalizing the results of \cite{DavDEMM} to the case of a non constant Robin parameter. In particular, we will prove the mutual absolute continuity of the Robin harmonic measure with respect to the surface measure in the setting of variable permeability.