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Dimension and structure of the Robin Harmonic Measure on Rough Domains

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arxiv 2410.23914 v1 pith:2OOXJYWM submitted 2024-10-31 math.AP math.CA

Guy David , Stefano Decio , Max Engelstein , Svitlana Mayboroda , Marco Michetti This is my paper · ORCID
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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.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Robin Green Function Estimates and a Model of Mammalian Lungs

    math.AP 2025-07 conditional novelty 7.0 of 10

    The paper proves a quantitative phase transition in the Robin Green function and shows a model lung maintains nearly constant oxygen transfer until permeability falls below a threshold set by the surface area.

  2. Robin harmonic measure with a variable permeability parameter

    math.AP 2025-10 conditional novelty 6.0 of 10

    Robin harmonic measure and the weighted surface measure a dσ are quantitatively mutually absolutely continuous on 1-sided NTA domains with d-ADR boundary for variable a in L^q.

  3. Equivalent Characterizations and Their Applications of Solvability of $L^p$ Poisson--Robin(-Regularity) Problems on Rough Domains

    math.AP 2025-07 conditional novelty 6.0 of 10

    The Lp Poisson-Robin and Poisson-Robin-regularity problems are equivalent (via duality and with the classical Robin plus Dirichlet problems), and for the Laplacian on Lipschitz domains they are solvable in sharp ranges of p.

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