The Eigenvalue Problem for the Laplacian via Conformal Mapping and the Gohberg--Sigal Theory
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866909354687463424 |
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| author | Beceanu, Marius Hong, Jiho Kwon, Hyun-Kyoung Lim, Mikyoung |
| author_facet | Beceanu, Marius Hong, Jiho Kwon, Hyun-Kyoung Lim, Mikyoung |
| contents | We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a planar, simply connected domain. The eigenvalues admit a characterization in terms of a layer potential of the Helmholtz equation. Using the exterior conformal mapping associated with the given domain, we reformulate the layer potential as an infinite-dimensional matrix. Based on this matrix representation, we develop a finite section approach for approximating the Laplacian eigenvalues and provide a convergence analysis by applying the Gohberg--Sigal theory for operator-valued functions. Moreover, we derive an asymptotic formula for the Laplacian eigenvalues on deformed domains that results from the changes in the conformal mapping coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_11026 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Eigenvalue Problem for the Laplacian via Conformal Mapping and the Gohberg--Sigal Theory Beceanu, Marius Hong, Jiho Kwon, Hyun-Kyoung Lim, Mikyoung Numerical Analysis Analysis of PDEs Spectral Theory 30C35, 35J05, 35P15, 41A25, 47A56 We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a planar, simply connected domain. The eigenvalues admit a characterization in terms of a layer potential of the Helmholtz equation. Using the exterior conformal mapping associated with the given domain, we reformulate the layer potential as an infinite-dimensional matrix. Based on this matrix representation, we develop a finite section approach for approximating the Laplacian eigenvalues and provide a convergence analysis by applying the Gohberg--Sigal theory for operator-valued functions. Moreover, we derive an asymptotic formula for the Laplacian eigenvalues on deformed domains that results from the changes in the conformal mapping coefficients. |
| title | The Eigenvalue Problem for the Laplacian via Conformal Mapping and the Gohberg--Sigal Theory |
| topic | Numerical Analysis Analysis of PDEs Spectral Theory 30C35, 35J05, 35P15, 41A25, 47A56 |
| url | https://arxiv.org/abs/2112.11026 |