Comparison inequalities for Dirichlet-to-Neumann maps

Denis S. Grebenkov, Michael Levitin, Karl-Mikael Perfekt, Iosif Polterovich

arXiv:2608.18678v1 Primary: math.SP Secondary: math.AP, math.FA

Abstract

We prove comparison inequalities for Dirichlet-to-Neumann maps corresponding to different non-positive Helmholtz parameters. For convex domains our bounds are sharp, and the resulting eigenvalue inequalities partially confirm an earlier conjecture, which we show does not hold in full generality. We further obtain geometry-dependent versions for arbitrary sufficiently regular domains, together with extensions to compact Riemannian manifolds with boundary. We also discuss analogous questions for metric graphs.

Disclosure

“AI usage disclosure The authors used different large language models, in particular ChatGPT 5.6 and Claude Opus 5, ex- tensively for mathematical discussions and editorial assistance. In particular, AI suggested that the ge- ometric properties of the distance function to the boundary should play a key role in the argument, and helped constructi”
PDF page 24

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-22T18:26:47.723062Z

Structural counts

Pages 25 pdf

Theorems 6 source

Lemmas 3 source

Propositions 3 source

Corollaries 6 source

Definitions 1 source

Displayed equations 140 source

Bibliography entries 27 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file GLPP-arXiv-v1.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.