The Levine--Weinberger and Friedlander--Filonov inequalities for some classes of elliptic operators

We consider the eigenvalue problem for certain classes of elliptic operators, namely inhomogeneous membrane operators $L=\frac{1}{\rho }(-\mathrm{\Delta }+V)$ and divergence form operators $L=-\mathrm{div}A\mathrm{\nabla }$, on bounded domains. For these operators, we prove ordering inequalities between the Dirichlet and the Neumann eigenvalues, generalizing results of Levine–Weinberger and Friedlander–Filonov for the Laplacian. We take inspiration from their proofs and derive sufficient conditions on the coefficients of the operator that ensure that the inequalities remain valid.