Shape perturbation of Grushin eigenvalues

TL;DR

This paper studies how the eigenvalues of the Grushin Laplacian change under domain perturbations, providing formulas for shape derivatives, bifurcation analysis, and characterizations of critical shapes.

Contribution

It establishes real analytic dependence of eigenvalues on domain shape, derives shape differential formulas, and characterizes critical shapes via overdetermined problems.

Findings

Eigenvalues depend real analytically on domain perturbations

Derived Hadamard-type formula for shape derivatives

Characterized bifurcation of multiple eigenvalues

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.