The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains
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 $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> -finite one-dimensional Hausdorff measure. For Reifenberg flat domains, this result extends a previous work of Thomas H. Wolff for the harmonic measure.
Authors
- Martí Prats (ORCID: https://orcid.org/0000-0001-8799-6995)
- Xavier Tolsa (ORCID: https://orcid.org/0000-0001-7976-5433)
- Ignasi Guillén-Mola (ORCID: https://orcid.org/0000-0002-1681-7002)
Institutions
- Institució Catalana de Recerca i Estudis Avançats (ES)
- Universitat Autònoma de Barcelona (ES)
- Institut Català de Ciències del Clima (ES)
- Centre de Recerca Matemàtica (ES)
Publication Details
- Journal
- Analysis and Mathematical Physics
- Published
- 2025-07-12
- DOI
- https://doi.org/10.1007/s13324-025-01067-5
- Primary Topic
- Advanced Harmonic Analysis Research
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- Centres de Recerca de Catalunya
- European Commission
- Deutsche Forschungsgemeinschaft
- Generalitat de Catalunya
- Universitat Autònoma de Barcelona
- Hausdorff Center for Mathematics
- Agència de Gestió d'Ajuts Universitaris i de Recerca
- Agencia Estatal de Investigación