On analysis of problems of mathematical physics with non-Lipschitz boundaries
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914017301233664 |
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| author | Rozanova-Pierrat, Anna |
| author_facet | Rozanova-Pierrat, Anna |
| contents | We review recent advances in solving problems of mathematical physics on domains with irregular boundaries in Rn. We distinguish two frameworks: a measure-free approach in the image of the trace operator spaces for extension domains and an L2-approach depending on a d-upper regular boundary measure. In both cases, the domains can have boundaries with different Hausdorff dimensions inside the interval (n -- 2, n). The generalization of the Poincar{é}-Steklov/Dirichlet-to-Neumann operator for these two contexts is given. To illustrate the established convergence of spectral problems for elliptic operators with Robin boundary conditions, we give a numerical example of the stability of localized eigenfunctions, using results of M. Graffin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On analysis of problems of mathematical physics with non-Lipschitz boundaries Rozanova-Pierrat, Anna Analysis of PDEs Functional Analysis We review recent advances in solving problems of mathematical physics on domains with irregular boundaries in Rn. We distinguish two frameworks: a measure-free approach in the image of the trace operator spaces for extension domains and an L2-approach depending on a d-upper regular boundary measure. In both cases, the domains can have boundaries with different Hausdorff dimensions inside the interval (n -- 2, n). The generalization of the Poincar{é}-Steklov/Dirichlet-to-Neumann operator for these two contexts is given. To illustrate the established convergence of spectral problems for elliptic operators with Robin boundary conditions, we give a numerical example of the stability of localized eigenfunctions, using results of M. Graffin. |
| title | On analysis of problems of mathematical physics with non-Lipschitz boundaries |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2509.02176 |