论文标题
Grushin特征值的形状扰动
Shape perturbation of Grushin eigenvalues
论文作者
论文摘要
我们认为，在$ \ mathbb {r}^n $的有限开放子集中，Grushin Laplacian的光谱问题受均匀的Dirichlet边界条件。我们证明特征值的对称函数在分析上取决于域扰动，并且我们证明了Hadamard-type公式用于其形状差异。在扰动的情况下，取决于单个标量参数，我们证明了Rellich-Nagy型定理，描述了多个特征值的分叉现象。作为推论，我们根据过度确定的问题来表征在等量和等值扰动下的临界形状，并推断出Grushin特征值的Rellich-Pohozaev身份的新证明。
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.
