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 = \tfrac{1}{ ρ} ( -Δ+ V ) $ and divergence form operators $ L = -\operatorname{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.