## 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
Full Text: