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

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arxiv 2406.11604 v2 pith:56H3ITXW submitted 2024-06-17 math.AP math.CA

Ignasi Guillén-Mola , Martí Prats , Xavier Tolsa This is my paper · ORCID
classification math.APmath.CA
keywords ellipticmeasureflatreifenbergdimensiondomainshausdorfflipschitz
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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 $\sigma$-finite one-dimensional Hausdorff measure. For Reifenberg flat domains, this result extends a previous work of Thomas H. Wolff for the harmonic measure.

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  1. Boundary regularity for the polyharmonic Dirichlet problem

    math.AP 2025-02 conditional novelty 7.0 of 10

    Weak solutions of the m-polyharmonic Dirichlet problem in Reifenberg-flat domains are C^{m-1,α} up to the boundary, with an a priori estimate in terms of the L^q norm of the source.

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