The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910922067410944 |
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| author | Guillén-Mola, Ignasi Prats, Martí Tolsa, Xavier |
| author_facet | Guillén-Mola, Ignasi Prats, Martí Tolsa, Xavier |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains Guillén-Mola, Ignasi Prats, Martí Tolsa, Xavier Analysis of PDEs Classical Analysis and ODEs 28A12, 31A15, 35J25. Secondary: 28A25, 28A78, 31B05, 35J08 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. |
| title | The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains |
| topic | Analysis of PDEs Classical Analysis and ODEs 28A12, 31A15, 35J25. Secondary: 28A25, 28A78, 31B05, 35J08 |
| url | https://arxiv.org/abs/2406.11604 |