Convergence of layer potentials and Riemann-Hilbert problem on extension domains
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| Format: | Preprint |
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2024
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| _version_ | 1866918165868445696 |
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| author | Claret, Gabriel Rozanova-Pierrat, Anna Teplyaev, Alexander |
| author_facet | Claret, Gabriel Rozanova-Pierrat, Anna Teplyaev, Alexander |
| contents | We prove the convergence of layer potential operators for the harmonic transmission problem over a sequence of converging two-sided extension domains. Consequently, the Neumann-Poincar{é} operators, Calder{ó}n projectors, and associated Neumann series converge in this setting. As a result, we generalize the notion of Cauchy integrals and, in a sense, of Hilbert transforms for a class of extension domains. Our approach relies on dyadic approximations of arbitrary open sets, considering convergence in terms of characteristic functions, Hausdorff distance, and compact sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_03767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of layer potentials and Riemann-Hilbert problem on extension domains Claret, Gabriel Rozanova-Pierrat, Anna Teplyaev, Alexander Analysis of PDEs Complex Variables Functional Analysis Operator Algebras We prove the convergence of layer potential operators for the harmonic transmission problem over a sequence of converging two-sided extension domains. Consequently, the Neumann-Poincar{é} operators, Calder{ó}n projectors, and associated Neumann series converge in this setting. As a result, we generalize the notion of Cauchy integrals and, in a sense, of Hilbert transforms for a class of extension domains. Our approach relies on dyadic approximations of arbitrary open sets, considering convergence in terms of characteristic functions, Hausdorff distance, and compact sets. |
| title | Convergence of layer potentials and Riemann-Hilbert problem on extension domains |
| topic | Analysis of PDEs Complex Variables Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2411.03767 |