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The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains

Ignasi Guillén-Mola, Martí Prats, Xavier Tolsa

TL;DR

This work extends Wolff's planar dimension bound for harmonic measure to elliptic measures associated with Lipschitz coefficient matrices in $(\delta,r_0)$‑Reifenberg flat domains with small $\delta$. The authors develop a chain of reductions, including linear changes of variables and Lipschitz diagonalization of the symmetric part of the coefficient matrix, to obtain a genus of quantifiable density estimates (Bourgain-type lemmas) and a $L\log L$ type control for the boundary density. A quantitative Main Lemma provides a full elliptic-measure cover of a boundary set with controlled total radius, which yields the existence of a full-measure boundary subset with σ‑finite 1-dimensional Hausdorff measure, hence $\dim_{\mathcal H} \omega_{\Omega,A}^p\le 1$. The results generalize Wolff's harmonic case to the elliptic setting under minimal boundary regularity, with explicit dependence on ellipticity and boundary flatness, and rely on Green’s function, Green- and Poisson-type representations, and a localization/covering strategy. Overall, the paper establishes a dimension drop phenomenon for planar elliptic measures in Reifenberg flat domains with Lipschitz coefficients, informing both boundary regularity theory and potential-theoretic properties of elliptic operators.

Abstract

In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff dimension of its elliptic measure is at most 1. More precisely, we prove that there exists a subset of the boundary with full elliptic measure and with $σ$-finite one-dimensional Hausdorff measure. For Reifenberg flat domains, this result extends a previous work of Thomas H. Wolff for the harmonic measure.

The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains

TL;DR

This work extends Wolff's planar dimension bound for harmonic measure to elliptic measures associated with Lipschitz coefficient matrices in ‑Reifenberg flat domains with small . The authors develop a chain of reductions, including linear changes of variables and Lipschitz diagonalization of the symmetric part of the coefficient matrix, to obtain a genus of quantifiable density estimates (Bourgain-type lemmas) and a type control for the boundary density. A quantitative Main Lemma provides a full elliptic-measure cover of a boundary set with controlled total radius, which yields the existence of a full-measure boundary subset with σ‑finite 1-dimensional Hausdorff measure, hence . The results generalize Wolff's harmonic case to the elliptic setting under minimal boundary regularity, with explicit dependence on ellipticity and boundary flatness, and rely on Green’s function, Green- and Poisson-type representations, and a localization/covering strategy. Overall, the paper establishes a dimension drop phenomenon for planar elliptic measures in Reifenberg flat domains with Lipschitz coefficients, informing both boundary regularity theory and potential-theoretic properties of elliptic operators.

Abstract

In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff dimension of its elliptic measure is at most 1. More precisely, we prove that there exists a subset of the boundary with full elliptic measure and with -finite one-dimensional Hausdorff measure. For Reifenberg flat domains, this result extends a previous work of Thomas H. Wolff for the harmonic measure.
Paper Structure (32 sections, 40 theorems, 322 equations, 2 figures)

This paper contains 32 sections, 40 theorems, 322 equations, 2 figures.

Table of Contents

  1. Introduction and Main results
  2. Preliminaries and definitions
  3. Notation
  4. CDC, NTA and Reifenberg flat domains
  5. Partial differential equations
  6. Non-degeneracy of the gradient of solutions of PDE's in Reifenberg flat domains with small constant
  7. The fundamental solution and the Green function
  8. Main Lemma and preliminary reductions
  9. The change of variables in Reduction 1
  10. Linear changes of variables
  11. Lipschitz diagonalization of symmetric matrices in the plane
  12. Reduction 1: Lipschitz diagonalization of the symmetric part
  13. Reduction 2: The dependence on the Lipschitz seminorm
  14. Elliptic measures in CDC domains
  15. Proof of the weak version of the Main Lemma
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\Omega \subset {\mathbb R}^2$ be a $(\delta, r_0)$-Reifenberg flat domain, $p\in \Omega$, and $A$ be a real uniformly elliptic (not necessarily symmetric) matrix with ellipticity constant $\lambda$. Suppose also that $A$ is Lipschitz. Then there exists $\delta_0 = \delta_0 (\lambda) >0$ such th

Figures (2)

  • Figure 1: Almost the worst situation: $\mathcal{P}\perp \{y=0\}$.
  • Figure 2: Setting of ().

Theorems & Definitions (112)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3: CDC domain. Heinonen2006
  • Definition 2.4: NTA domain. Jerison1982
  • Remark 2.5
  • Definition 2.6: Hausdorff distance
  • Definition 2.7: Reifenberg flat domain
  • Definition 2.8
  • Theorem 2.9: See Gilbarg2001
  • ...and 102 more