Spectrum of the Laplacian with weights
  Given a compact Riemannian manifold (M, g) and two positive functions $\rho$ and $\sigma$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $\sigma$, with respect to the L 2 inner product weighted by $\rho$. Under some regularity conditions on $\rho$ and $\sigma$, these eigenvalues are those of the operator $\rho$^{-1} div($\sigma$$\nabla$u) with Neumann conditions on the boundary if $\partial$M = $\emptyset$. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved. 
Bruno Colbois, Ahmad El Soufi (LMPT)
math.DG math.MG