On the estimates of eigenvalues of the boundary value problem with large parameter

Abstract

We consider the eigenvalue problem $\Delta u + \Lambda u = 0$ in $\Omega$ with Robin condition  $\frac {\delta u}{\delta \nu} + \alpha u = 0$ on $\delta\Omega$, where $\Omega \subset \R^n, n\geq 2$ is a bounded domain with  $\delta\Omega\in  C^2, \alpha$ is a real parameter. We obtain the estimates to the difference \lambda^D_k - \lamda_k(\alpha) between $k$-th eigenvalue of the Laplace operator in \Omega with Dirichlet condition and the corresponding Robin eigenvalue for large positive values of \alpha for all $k= 1,2,\ldots$ We also show sharpness of these estimates in the pover of $\alpha$.