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Robin harmonic measure with a variable permeability parameter

Svitlana Mayboroda, Alberto Pacati

TL;DR

This work analyzes Robin boundary value problems for elliptic operators with a variable, nonnegative permeability parameter on bounded 1‑sided NTA domains with $d$‑Ahlfors regular boundaries. It extends the constant Robin parameter theory to $a\in L^q(\partial\Omega)$ and proves mutual absolute continuity between the Robin harmonic measure and the boundary measure $\mu=a\,d\sigma$, with precise small and large scale estimates. The authors develop boundary-adapted Poincaré inequalities, Neumann techniques, and a Green function/Harmonic measure framework to obtain Hölder continuity, Harnack inequalities, and representation formulas for Robin solutions. These results unify the Dirichlet/Neumann regimes, provide robust quantitative tools for Robin problems in rough domains, and generalize DavDEMM to variable permeability.

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.

Robin harmonic measure with a variable permeability parameter

TL;DR

This work analyzes Robin boundary value problems for elliptic operators with a variable, nonnegative permeability parameter on bounded 1‑sided NTA domains with ‑Ahlfors regular boundaries. It extends the constant Robin parameter theory to and proves mutual absolute continuity between the Robin harmonic measure and the boundary measure , with precise small and large scale estimates. The authors develop boundary-adapted Poincaré inequalities, Neumann techniques, and a Green function/Harmonic measure framework to obtain Hölder continuity, Harnack inequalities, and representation formulas for Robin solutions. These results unify the Dirichlet/Neumann regimes, provide robust quantitative tools for Robin problems in rough domains, and generalize DavDEMM to variable permeability.

Abstract

In this paper we study the behavior of the solutions to the Robin problem in bounded -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.
Paper Structure (6 sections, 25 theorems, 136 equations)

This paper contains 6 sections, 25 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Set up and Poincaré inequalities
  3. Existence and uniqueness of solutions
  4. The Neumann problem
  5. Boundary Hölder continuity for Robin solutions
  6. Harmonic measure and Green functions

Key Result

Theorem 1.2

Let $\Omega\subset\mathbb{R}^n$ be a bounded one sided NTA domain, and let $\sigma$ be a $d$-Ahlfors regular measure supported in $\partial\Omega$, where $n-2<d<n$. Let $0\le a\in L^q(\partial\Omega,\sigma)$ be non identically zero, for some $q>\frac{d}{d-n+2}$. Then, for any $x\in \Omega$, we have where $d\mu=a\,d\sigma$ (see mu), and $\omega^X_{R,L}$ is the Robin harmonic measure related to the

Theorems & Definitions (52)

  • Theorem 1.2: Mutual absolute continuity
  • Theorem 1.3: Quantitative mutual absolute continuity at small scales
  • Theorem 1.5: Quantitative mutual absolute continuity at large scales
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.5
  • Theorem 2.11: Poincaré inequality
  • Definition 2.13
  • Proposition 2.17
  • Theorem 2.24
  • ...and 42 more