Hadamard-type variation formulas for the eigenvalues of the j-Laplacian and applications

Author(s)

Gomes, Jose Nazareno Vieira ; Marrocos, Marcus Antonio Mendonca ; Mesquita, Raul Rabello

• Document type: Journal article
• Source: JOURNAL OF SPECTRAL THEORY; v. 14, n. 4, p. 17-pg., 2024-01-01.

Abstract

We consider an analytic family of Riemannian metrics on a compact smooth manifold M. We assume the Dirichlet boundary condition for the r7-Laplacian and obtain Hadamardtype variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all C (R) Riemannian metrics M (R) on M, all eigenvalues of the r7Laplacian are generically simple, for 2 < r < infinity. This implies the existence of a residual set of metrics in M (R) that makes the spectrum of the r7-Laplacian simple. Likewise, we show that there exists a residual set of drifting functions r(7) in the space F (R) of all C (R) functions on M, that again makes the spectrum of the r7-Laplacian simple, for 2 < r < oo. 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 r7-Laplacian in this case. We also establish generic properties of eigenvalues in this context. (AU)

• FAPESP's process: 20/14075-6 - Dynamical systems and their attractors under perturbations
• Grantee: Alexandre Nolasco de Carvalho
• Support Opportunities: Research Projects - Thematic Grants