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

Abstract

We consider the eigenvalue problem for certain classes of elliptic operators, namely inhomogeneous membrane operators $L=\frac{1}{\rho }(-\Delta +V)$ and divergence form operators $L=-\mathrm{div}A\nabla$, 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.

minor modifications