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.