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On the eigenvalues of a Robin problem with a large parameter. (English) Zbl 1340.35238
Summary: We consider the Robin eigenvalue problem \(\Delta u+\lambda u=0\) in \(\Omega \), \({\partial u}/{\partial \nu}+\alpha u=0\) on \(\partial \Omega \) where \(\Omega \subset \mathbb R^n\), \(n \geqslant 2\) is a bounded domain and \(\alpha \) is a real parameter. We investigate the behavior of the eigenvalues \(\lambda_k (\alpha)\) of this problem as functions of the parameter \(\alpha \). We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative \(\lambda_1'(\alpha)\). Assuming that the boundary \(\partial \Omega \) is of class \(C^2\) we obtain estimates to the difference \(\lambda_k^D-\lambda_k(\alpha)\) between the \(k\)-th eigenvalue of the Laplace operator with Dirichlet boundary condition in \(\Omega \) and the corresponding Robin eigenvalue for positive values of \(\alpha \) for every \(k=1,2,\dots \).

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