Hadamard-type variation formulas for the eigenvalues of the $\eta$-Laplacian and applications

Abstract

We consider an analytic family of Riemannian metrics on a compact smooth manifold $M$. We assume the Dirichlet boundary condition for the $\eta$-Laplacian and obtain Hadamard-type variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all $C^r$ Riemannian metrics ${\mathcal{M}}^r$ on $M$, all eigenvalues of the $\eta$-Laplacian are generically simple, for $2\le r<\infty$. This implies the existence of a residual set of metrics in ${\mathcal{M}}^r$ that makes the spectrum of the $\eta$-Laplacian simple. Likewise, we show that there exists a residual set of drifting functions $\eta$ in the space ${\mathcal{F}}^r$ of all $C^r$ functions on $M$, that again makes the spectrum of the $\eta$-Laplacian simple, for $2\le r<\infty$. Besides, we provide a precise information about the complement of these residual sets as well as about the structure of the set of deformations of a Riemannian metric (respectively, of the set of deformations of a drifting function) which preserves double eigenvalues. Moreover, we consider a family of perturbations of a domain in a Riemannian manifold and obtain Hadamard-type formulas for the eigenvalues of the $\eta$-Laplacian in this case. We also establish generic properties of eigenvalues in this context.