  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.
