Generalization of Hadamard’s Laplace eigenvalue formula to deforming manifolds. (English) Zbl 1181.35148

Summary: For a Euclidean domain with a moving boundary, Hadamard’s formula relates the rate of change of the Laplace eigenvalues to the normal velocity of the boundary. We generalize Hadamard’s formula to deforming Riemannian manifolds with contour boundary moving in a compatible manner. Our analysis finds direct applications in the dynamics of fluid films. The spectrum of the surface Laplacian describes the frequencies of normal oscillations of the film’s surface as well as tangential oscillations in thickness.


35P05 General topics in linear spectral theory for PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
58J90 Applications of PDEs on manifolds
76A20 Thin fluid films