A maximum principle in spectral optimization problems for elliptic operators subject to mass density perturbations

We consider eigenvalue problems for general elliptic operators of arbitrary order subject to homogeneous boundary conditions on open subsets of the Euclidean $N$-dimensional space. We prove stability results for the dependence of the eigenvalues upon variation of the mass density and we prove a maximum principle for extremum problems related to mass density perturbations which preserve the total mass.