Behavior of Absorbing and Generating $p$-Robin Eigenvalues in Bounded and Exterior Domains
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917975348477952 |
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| author | Bundrock, Lukas Giorgi, Tiziana Smits, Robert |
| author_facet | Bundrock, Lukas Giorgi, Tiziana Smits, Robert |
| contents | We establish rigorous quantitative inequalities for the first eigenvalue of the generalized $p$-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter $α$ is positive, and the superconducting generation regime ($α<0$), where the boundary acts as a source. In bounded domains, we use a unified approach to derive a precise asymptotic behavior for all $p$ and all small real $α$, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem, studied in the fundamental work by René Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as $α\to 0$ for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06236 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Behavior of Absorbing and Generating $p$-Robin Eigenvalues in Bounded and Exterior Domains Bundrock, Lukas Giorgi, Tiziana Smits, Robert Analysis of PDEs Spectral Theory We establish rigorous quantitative inequalities for the first eigenvalue of the generalized $p$-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter $α$ is positive, and the superconducting generation regime ($α<0$), where the boundary acts as a source. In bounded domains, we use a unified approach to derive a precise asymptotic behavior for all $p$ and all small real $α$, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem, studied in the fundamental work by René Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as $α\to 0$ for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions. |
| title | Behavior of Absorbing and Generating $p$-Robin Eigenvalues in Bounded and Exterior Domains |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2408.06236 |