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Estimates of eigenvalues of a boundary value problem with a parameter. (English) Zbl 1323.35114

The author considers the eigenvalue problem \[ \begin{aligned} \Delta u+\lambda u =0\quad &\text{in }\Omega, \\ \frac{\partial u}{\partial\nu}+\alpha\sigma(x)u=0\quad &\text{in }\partial\Omega,\end{aligned} \] where \(\Omega\subset\mathbb R^n\) is a bounded domain with boundary \(\partial\Omega\in C^2\), \(\nu\) denotes the outward unit normal vector to \(\partial\Omega\), \(\alpha\) is a real parameter, and the function \(\sigma\in C^1(\partial\Omega)\) is bounded and positive. There is a sequence of eigenvalues \(\lambda_1(\alpha)< \lambda_2(\alpha) \leq\cdots\) of this problem. Consider also the sequence of eigenvalues \(0< \lambda^D_1< \lambda^D_2\leq\cdots\) of the Dirichlet Laplacian. The author estimates \(\lambda_k(\alpha)\) for large values of \(\alpha\). The main result of this paper is that \[ 0\leq\lambda^D_k-\lambda_k(\alpha)\leq C_1\alpha^{-1/2}\left(\lambda^D_k\right)^2, \] where the constant \(C_1\) does not depend on \(k\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
49R05 Variational methods for eigenvalues of operators
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