Spectral asymptotics for Robin laplacians on Lipschitz sets

Simon Larson^a, Rupert L. Frank<sup>bcd

a</sup> Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden
^b Mathematisches Institut, Ludwig-Maximilians-Universität München, München, Germany
^c Munich Center for Quantum Science and Technology, München, Germany
^d California Institute of Technology, Mathematics 253-37, Pasadena, USA

Abstract: We prove two-term spectral asymptotics for the Riesz means of the eigenvalues of the Laplacian on a Lipschitz domain with Robin boundary conditions. The second term is the same as in the case of Neumann boundary conditions. This is valid for Riesz means of arbitrary positive order. For orders at least one and under additional assumptions on the function determining the boundary conditions, we derive leading order asymptotics for the difference between Riesz means of Robin and Neumann eigenvalues.

Keywords: semiclassical asymptotics, Laplacian, Robin boundary conditions.

MSC: 35P20, 35P15, 35J15

Received: 29.10.2024
Revised: 14.11.2024
Accepted: 18.11.2024

DOI: 10.4213/faa4266

 English version: Functional Analysis and Its Applications, 2025, 59:3, 277–296

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