Work

Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains

Year: 2025

Type: article

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 , extend to certain values , 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.

Source: Communications on Pure and Applied Mathematics

DOI: 10.1002/cpa.70019

Authors: Rupert L. Frank, Simon Larson

Institutions: California Institute of Technology, Munich Center for Quantum Science and Technology, Ludwig-Maximilians-Universität München, Chalmers University of Technology, University of Gothenburg

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $5,230

Funders: National Science Foundation, Deutsche Forschungsgemeinschaft, Knut och Alice Wallenbergs Stiftelse, Vetenskapsrådet