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Dimension and structure of the Robin Harmonic Measure on Rough Domains

Guy David, Stefano Decio, Max Engelstein, Svitlana Mayboroda, Marco Michetti

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.

Dimension and structure of the Robin Harmonic Measure on Rough Domains

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.

Paper Structure

This paper contains 14 sections, 31 theorems, 212 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometric assumptions
  3. Definition of the Robin elliptic measure and fundamental regularity problems of solutions to the Robin problem
  4. Absolute continuity of the Robin harmonic measure
  5. Outline of the Paper and Comparison with Other Work
  6. Functional Analytic Preliminaries and Poincaré
  7. An existence result for Robin boundary data
  8. A localization lemma
  9. More Poincaré inequalities
  10. Partial Neumann Data
  11. Hölder Continuity at the Boundary for the Neumann problem
  12. Hölder continuity for the Robin problem
  13. Existence of Robin Harmonic Measure
  14. Absolute continuity of the Robin Harmonic Measure

Key Result

Theorem 1.1

Let $(\Omega, \sigma)$ be a one-sided NTA pair of mixed dimension, with $\Omega$ bounded, and let $A$ be a uniformly elliptic real matrix valued function. Let $u\in W^{1,2}(\Omega)$ satisfy weakform for some $f\in C^{0,\beta}(\partial \Omega)$. There exists an $\alpha \in (0,1)$, which depends only

Figures (2)

  • Figure 1: First three steps of the construction of the complement of the four corner Cantor set
  • Figure 2: First three steps of the construction of a Koch snowflake

Theorems & Definitions (71)

  • Definition 1
  • Example 1
  • Example 2
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Remark 1
  • ...and 61 more