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 $\mathcal{M}^r$, which 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 $M$, which makes again the spectrum of the $\eta$-Laplacian simple, for $2\leq r< \infty$. Besides, we give a precise information about the complementary 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.