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Laplace operator; Robin boundary condition; eigenvalue; large parameter
We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega $, ${\partial u}/{\partial \nu }+\alpha u=0$ on $\partial \Omega $ where $\Omega \subset \mathbb R^n$, $n \geq 2$ 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 $.
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