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On the asymptotic behaviour of the eigenvalues of a Robin problem. (English) Zbl 1240.35370
The authors prove that every eigenvalue of a Robin problem \(-\Delta u=\lambda u\) in \(\Omega \), \(\frac {\partial u}{\partial \nu }=\alpha u\) on \(\partial \Omega ,\) where \(\alpha \) is a positive boundary parameter and \(\Omega \subset \mathbb {R}^{n}\) is a bounded domain of class \(C^1,\) behaves asymptotically like \(-\alpha ^2\) as \(\alpha \rightarrow \infty \). This generalizes an existing result for the first eigenvalue.

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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