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

2025/06/11 by T. Schmatzler, Schmatzler, T.
Mathematics · #Spectral Theory in Mathematical Physics #Nonlinear Partial Differential Equations #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2506.09468

Abstract

We consider the eigenvalue problem for certain classes of elliptic operators, namely inhomogeneous membrane operators L = \tfrac1 ρ ( -Δ+ V ) and divergence form operators L = -div A ∇ , 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.

Citations

• Inequalities between Dirichlet and Neumann eigenvalues on Carnot groups
• Inequalities between Dirichlet and Neumann eigenvalues of the magnetic Laplacian
• Inequalities between Neumann and Dirichlet Laplacian eigenvalues on planar domains
• Inequalities between Neumann and Dirichlet eigenvalues of Schrödinger operators
• Isoperimetric relations between Dirichlet and Neumann eigenvalues
• Eigenvalue inequalities for the Laplacian with mixed boundary conditions

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