Steklov eigenvalues of nearly hyperspherical domains
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908547399286784 |
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| author | Tan, Chee Han Viator, Robert |
| author_facet | Tan, Chee Han Viator, Robert |
| contents | We consider Steklov eigenvalues of nearly hyperspherical domains in $\mathbb{R}^{d + 1}$ with $d\ge 3$. In previous work, treating such domains as perturbations of the ball, we proved that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion and show that the first-order perturbations are eigenvalues of a Hermitian matrix, whose entries can be written explicitly in terms of the Pochhammer's and Wigner $3j$-symbols. We analyse the asymptotic expansion and show the following isoperimetric results among domains with fixed volume: (1) for an infinite subset of Steklov eigenvalues, the ball is not optimal, and (2) for a different infinite subset of Steklov eigenvalues, the ball is a stationary point. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_03960 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Steklov eigenvalues of nearly hyperspherical domains Tan, Chee Han Viator, Robert Spectral Theory Analysis of PDEs Optimization and Control 35C20, 35P05, 41A58 We consider Steklov eigenvalues of nearly hyperspherical domains in $\mathbb{R}^{d + 1}$ with $d\ge 3$. In previous work, treating such domains as perturbations of the ball, we proved that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion and show that the first-order perturbations are eigenvalues of a Hermitian matrix, whose entries can be written explicitly in terms of the Pochhammer's and Wigner $3j$-symbols. We analyse the asymptotic expansion and show the following isoperimetric results among domains with fixed volume: (1) for an infinite subset of Steklov eigenvalues, the ball is not optimal, and (2) for a different infinite subset of Steklov eigenvalues, the ball is a stationary point. |
| title | Steklov eigenvalues of nearly hyperspherical domains |
| topic | Spectral Theory Analysis of PDEs Optimization and Control 35C20, 35P05, 41A58 |
| url | https://arxiv.org/abs/2310.03960 |