## 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

### Keywords:

eigenvalue; Robin problem; Laplacian
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