Hadamard-type variation formulae for the eigenvalues of a class of second-order elliptic operators and its applications
Abstract
Abstract We use variational methods to derive Hadamard-type formulae for the eigenvalues of a class of elliptic operators on a compact Riemannian manifold M . We then apply the latter in the following context. Consider a family of elliptic operators which is parametrized by either the set of all $$\mathcal {C}^r$$ C r –Riemannian metrics on M or the set of all $$\mathcal {C}^r$$ C r –diffeomorphisms on a domain into M . In either case, we prove that if a subset of the parametrizations set yields a simple spectrum of the operator, then it is necessarily a generic subset. We also analyse the behavior of the eigenvalues when the metric evolves along the Ricci flow on a closed Riemannian manifold, and we prove, under a suitable hypothesis, that they increase.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Nonlinear Partial Differential Equations
Analytic and geometric function theory
Publication details
- Journal / source
- Nonlinear Differential Equations and Applications NoDEA
- Publisher
- Springer Science and Business Media LLC
- Volume
- 33
- Issue
- 5
- Publication date
- 16 July 2026
- Language
- EN