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

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\leq 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\leq 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.