Mean curvature bounds and eigenvalues of Robin Laplacians
| hal.structure.identifier | Laboratoire de Mathématiques d'Orsay [LMO] | |
| dc.contributor.author | PANKRASHKIN, Konstantin | |
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| hal.structure.identifier | Centre de Physique Théorique - UMR 7332 [CPT] | |
| dc.contributor.author | POPOFF, Nicolas | |
| dc.date.accessioned | 2024-04-04T03:21:55Z | |
| dc.date.available | 2024-04-04T03:21:55Z | |
| dc.date.created | 2014-07-11 | |
| dc.date.issued | 2015 | |
| dc.identifier.issn | 0944-2669 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/194703 | |
| dc.description.abstractEn | We consider the Laplacian with attractive Robin boundary conditions, \[ Q^\Omega_\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text{ on } \partial\Omega, \] in a class of bounded smooth domains $\Omega \in\mathbb{R}^\nu$; here $n$ is the outward unit normal and $\alpha>0$ is a constant. We show that for each $j\in\mathbb{N}$ and $\alpha\to+\infty$, the $j$th eigenvalue $E_j(Q^\Omega_\alpha)$ has the asymptotics \[ E_j(Q^\Omega_\alpha)=-\alpha^2 -(\nu-1)H_\mathrm{max}(\Omega)\,\alpha+{\mathcal O}(\alpha^{2/3}), \] where $H_\mathrm{max}(\Omega)$ is the maximum mean curvature at $\partial \Omega$. The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of $H_\mathrm{max}$. In particular, we show that the ball is the strict minimizer of $H_\mathrm{max}$ among the smooth star-shaped domains of a given volume, which leads to the following result: if $B$ is a ball and $\Omega$ is any other star-shaped smooth domain of the same volume, then for any fixed $j\in\mathbb{N}$ we have $E_j(Q^B_\alpha)>E_j(Q^\Omega_\alpha)$ for large $\alpha$. An open question concerning a larger class of domains is formulated. | |
| dc.language.iso | en | |
| dc.publisher | Springer Verlag | |
| dc.title.en | Mean curvature bounds and eigenvalues of Robin Laplacians | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.1007/s00526-015-0850-1 | |
| dc.subject.hal | Mathématiques [math]/Equations aux dérivées partielles [math.AP] | |
| dc.subject.hal | Mathématiques [math]/Théorie spectrale [math.SP] | |
| dc.subject.hal | Mathématiques [math]/Géométrie différentielle [math.DG] | |
| dc.subject.hal | Mathématiques [math]/Optimisation et contrôle [math.OC] | |
| dc.subject.hal | Mathématiques [math]/Physique mathématique [math-ph] | |
| bordeaux.journal | Calculus of Variations and Partial Differential Equations | |
| bordeaux.page | 1947-1961 | |
| bordeaux.volume | 54 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-01022945 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Non spécifiée | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01022945v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Calculus%20of%20Variations%20and%20Partial%20Differential%20Equations&rft.date=2015&rft.volume=54&rft.spage=1947-1961&rft.epage=1947-1961&rft.eissn=0944-2669&rft.issn=0944-2669&rft.au=PANKRASHKIN,%20Konstantin&POPOFF,%20Nicolas&rft.genre=article |
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