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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2408.06236 |
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- 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.