On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
We study the limit as p goes to infinity of the first non-zero eigenvalue λp of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U of Rn. We prove that λ∞:=lim λp1/p=2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can think...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2016 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/59878 |
| Acceso en línea: | http://hdl.handle.net/11336/59878 |
| Access Level: | acceso abierto |
| Palabra clave: | infinity laplacian eigenvalue shape derivative neumann boundary condition https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| Sumario: | We study the limit as p goes to infinity of the first non-zero eigenvalue λp of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U of Rn. We prove that λ∞:=lim λp1/p=2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can think of λ∞ as the first eigenvalue of the infinity-Laplacian with Neumann boundary conditions. We also study the regularity of λ∞ as a function of the domain U proving that, under a smooth perturbation Ut of U by diffeomorphisms close to the identity, there holds that λ∞(Ut)=λ∞(U)+O(t). Although λ∞(Ut) is in general not differentiable at t=0, we provide sufficient geometric conditions for its differentiability with an explicit formula for the derivative. |
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