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
Full text available as PDF.
Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.