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

TL;DR

This paper extends classical eigenvalue inequalities to specific classes of elliptic operators, providing conditions under which Dirichlet and Neumann eigenvalues are ordered, generalizing known results for the Laplacian.

Contribution

It introduces new ordering inequalities for eigenvalues of inhomogeneous membrane and divergence form elliptic operators, generalizing Levine--Weinberger and Friedlander--Filonov results.

Findings

Established eigenvalue inequalities for inhomogeneous membrane operators.

Derived conditions on coefficients ensuring eigenvalue ordering.

Generalized classical inequalities to broader classes of elliptic operators.

Abstract

We consider the eigenvalue problem for certain classes of elliptic operators, namely inhomogeneous membrane operators $ L = \tfrac{1}{ \rho } ( -\Delta + 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.