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Shape Perturbation of Grushin Eigenvalues
The Journal of Geometric Analysis ( IF 1.1 ) Pub Date : 2021-04-09 , DOI: 10.1007/s12220-021-00662-9
Pier Domenico Lamberti , Paolo Luzzini , Paolo Musolino

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.



中文翻译:

Grushin特征值的形状扰动

我们考虑\({\ mathbb {R}} ^ N \的有界开放子集上的均质Dirichlet边界条件下的Grushin Laplacian谱问题。我们证明了特征值的对称函数真正取决于解析域扰动,并且我们证明了其形状微分的Hadamard型公式。在取决于单个标量参数的摄动情况下,我们证明了描述多个特征值分叉现象的Rellich-Nagy型定理。作为推论,我们用超定问题来描述等体积和等体积扰动下的临界形状,并为格鲁什特征值推导了Rellich-Pohozaev身份的新证明。

更新日期:2021-04-09
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