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Mathematics > Spectral Theory

arXiv:2412.03369 (math)
[Submitted on 4 Dec 2024]

Title:Refining Friedlander-Filonov inequality with the de Rham complex

Authors:Magnus Fries, Magnus Goffeng, Germán Miranda
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Abstract:Inequalities between Dirichlet and Neumann eigenvalues of the Laplacian and of other differential operators have been intensively studied in the past decades. The aim of this paper is to introduce differential forms and the de Rham complex in the study of such inequalities. We show how differential forms lie hidden at the heart of work of Rohleder on inequalities between Dirichlet and Neumann eigenvalues for the Laplacian on planar domains. Moreover, we extend and generalize the result of Rohleder by proving that the j+2:nd eigenvalue of the Neumann Laplacian is less than or equal to the j:th eigenvalue of the Dirichlet Laplacian for any compact Lipschitz domain in dimension two or greater.
Comments: 15 pages
Subjects: Spectral Theory (math.SP); Analysis of PDEs (math.AP); Differential Geometry (math.DG)
MSC classes: 35P05
Cite as: arXiv:2412.03369 [math.SP]
  (or arXiv:2412.03369v1 [math.SP] for this version)
  https://doi.org/10.48550/arXiv.2412.03369

Submission history

From: Germán Miranda [view email]
[v1] Wed, 4 Dec 2024 14:57:52 UTC (18 KB)
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