Geodesic
Spectral Stability of the Robin Laplacian
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 75-96.

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We consider the Robin Laplacian in two bounded regions $\Omega_1$ and $\Omega_2$ of $\mathbb R^N$ with Lipschitz boundaries and such that $\Omega_2\subset\Omega_1$, and we obtain two-sided estimates for the eigenvalues $\lambda_{n,2}$ of the Robin Laplacian in $\Omega_2$ via the eigenvalues $\lambda_{n,1}$ of the Robin Laplacian in $\Omega_1$. Our estimates depend on the measure of the set difference $\Omega_1\!\setminus\Omega_2$ and on suitably defined characteristics of vicinity of the boundaries $\partial\Omega_1$ and $\partial\Omega_2$, and of the functions defined on $\partial\Omega_1$ and on $\partial\Omega_2$ that enter the Robin boundary conditions. 
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V. I. Burenkov; M. Lanza de Cristoforis. Spectral Stability of the Robin Laplacian. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 75-96. http://geodesic.mathdoc.fr/item/TM_2008_260_a5/

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