Shape perturbation of Grushin eigenvalues

Pier Domenico Lamberti; Paolo Luzzini; Paolo Musolino

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arXiv · 2009.03130

Shape perturbation of Grushin eigenvalues

Pier Domenico Lamberti·Paolo Luzzini·Paolo Musolino

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Abstract

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $\mathbb{R}^N$. We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich-Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich-Pohozaev identity for the Grushin eigenvalues.

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Pier Domenico Lamberti, Paolo Luzzini, Paolo Musolino. 2020-09-07. Shape perturbation of Grushin eigenvalues. https://arxiv.org/abs/2009.03130

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