Filinovskiy Alexey, Department of High Mathematics, Faculty of Fundamental Sciences, Moscow State Technical University, Moskva, 2-nd Baumanskaya ul. 5, 105005, Russian Federation, e-mail: flnv@yandex.ru
Abstract: We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega$, ${\partial u}/{\partial\nu}+\alpha u=0$ on $\partial\Omega$ where $\Omega\subset\mathbb R^n$, $n \geq2$ 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 $k=1,2,\dots$.
Keywords: Laplace operator; Robin boundary condition; eigenvalue; large parameter
Classification (MSC 2010): 35P15, 35J05
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