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Published October 24, 2025 | Version Early View
Journal Article Open

Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains

Creators

  • 1. ROR icon Ludwig-Maximilians-Universität München
  • 2. ROR icon California Institute of Technology
  • 3. ROR icon University of Gothenburg

Abstract

We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that the classical inequalities of Berezin–Li–Yau and Kröger, valid for Riesz exponents γ ≥ 1, extend to certain values γ < 1 , provided the underlying domain is convex. We also study the corresponding optimization problems and describe the implications of a possible failure of Pólya's conjecture for convex sets in terms of Riesz means. These findings allow us to describe the asymptotic behavior of solutions of a spectral shape optimization problem for convex sets.

Copyright and License

© 2025 The Author(s). Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Funding

  • National Science Foundation. Grant Number: DMS-1954995
  • German Research Foundation. Grant Numbers: EXC-2111-390814868, TRR 352-Project-ID 470903074
  • Knut and Alice Wallenberg Foundation. Grant Number: KAW 2017.0295
  • Swedish Research Council. Grant Number: 2023-03985

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Additional details

Funding

National Science Foundation
DMS‐1954995
Deutsche Forschungsgemeinschaft
EXC‐2111‐390814868
Deutsche Forschungsgemeinschaft
TRR 352‐Project‐ID 470903074
Knut and Alice Wallenberg Foundation
KAW 2017.0295
Swedish Research Council
2023‐03985

Dates

Accepted
2025-09-08
Accepted
Available
2025-10-24
Version of record

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Caltech groups
Division of Physics, Mathematics and Astronomy (PMA)
Publication Status
In Press