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arXiv · 2406.11604

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

Ignasi Guillén-Mola·Martí Prats·Xavier Tolsa
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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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Ignasi Guillén-Mola, Martí Prats, Xavier Tolsa. 2024-06-17. The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains. https://doi.org/10.1007/s13324-025-01067-5

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