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The Singular Strata of a Free-Boundary problem for harmonic measure

Sean McCurdy

Abstract

In this paper, we obtain \textit{quantitative} estimates on the fine structure of the singular set of the mutual boundary $\partial Ω^{\pm}$ for pairs of complementary domains, $Ω^+, Ω^- \subset \mathbb{R}^n$ which arise in a class of two-sided free boundary problems for harmonic measure. These estimates give new insight into the structure of the mutual boundary, $\partial Ω^{\pm}.$

The Singular Strata of a Free-Boundary problem for harmonic measure

Abstract

In this paper, we obtain \textit{quantitative} estimates on the fine structure of the singular set of the mutual boundary for pairs of complementary domains, which arise in a class of two-sided free boundary problems for harmonic measure. These estimates give new insight into the structure of the mutual boundary,

Paper Structure

This paper contains 25 sections, 49 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Definitions and Statement of Main Results
  3. Domains and their Green's functions.
  4. A class of functions and their rescalings
  5. Quantitative symmetry.
  6. Main Results and Outline of the Proof
  7. Outline of the Proof of Theorem 3.1
  8. Compactness
  9. Almost monotonicity of the Almgren frequency function
  10. Uniform Bound on the Almgren frequency
  11. Quantitative Rigidity
  12. A Dichotomy
  13. Spatial Derivatives of the Almgren Frequency
  14. Frequency Pinching
  15. The proof of Lemma \ref{['beta bound lem']}
  16. ...and 10 more sections

Key Result

Theorem 2.8

(KenigToro06, Badger11, Engelstein16) For $v^Q_{Q, r}$ and $\omega^{\pm}_{Q, r}$ as in Definition harmonic rescaling def,

Theorems & Definitions (112)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 102 more