Dimension and structure of the Robin Harmonic Measure on Rough Domains

TL;DR

This paper proves that Robin harmonic measure remains mutually absolutely continuous with surface measure on rough domains, contradicting previous expectations of a phase transition in its dimension as boundary conditions vary.

Contribution

It establishes quantitative mutual absolute continuity of Robin harmonic measure with surface measure on rough domains, showing no dimension drop and no need for boundary rectifiability.

Findings

Robin harmonic measure is mutually absolutely continuous with surface measure.

No dimension drop occurs in Robin harmonic measure on rough domains.

The phase transition in measure dimension is not observed as previously expected.

Abstract

The present paper establishes that the Robin harmonic measure is quantitatively mutually absolutely continuous with respect to the surface measure on any Ahlfors regular set in any (quantifiably) connected domain for any elliptic operator. This stands in contrast with analogous results for the Dirichlet boundary value problem and also contradicts the expectation, supported by simulations in the physics literature, that the dimension of the Robin harmonic measure in rough domains exhibits a phase transition as the boundary condition interpolates between completely reflecting and completely absorbing. In the adopted traditional language, the corresponding harmonic measure exhibits no dimension drop, and the absolute continuity necessitates neither rectifiability of the boundary nor control of the oscillations of the coefficients of the equation. The expected phase transition is rather exhibited through the detailed non-scale-invariant weight estimates.