A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

Stefano Biagi; Serena Dipierro; Enrico Valdinoci; Eugenio Vecchi; Stefano Biagi; Serena Dipierro; Enrico Valdinoci; Eugenio Vecchi

doi:10.3934/mine.2023014

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-25. doi: 10.3934/mine.2023014

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A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

1.
Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
2.
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA 6009 Crawley, Australia
3.
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

Received: 14 October 2021 Revised: 26 December 2021 Accepted: 18 January 2022 Published: 28 February 2022

Given a bounded open set $ \Omega\subseteq{\mathbb{R}}^n $, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $ \Omega $. We prove that the second eigenvalue $ \lambda_2(\Omega) $ is always strictly larger than the first eigenvalue $ \lambda_1(B) $ of a ball $ B $ with volume half of that of $ \Omega $. This bound is proven to be sharp, by comparing to the limit case in which $ \Omega $ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.
- operators of mixed order,
- first eigenvalue,
- shape optimization,
- isoperimetric inequality,
- Faber-Krahn inequality,
- quantitative results,
- stability
Citation: Stefano Biagi, Serena Dipierro, Enrico Valdinoci, Eugenio Vecchi. A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators[J]. Mathematics in Engineering, 2023, 5(1): 1-25. doi: 10.3934/mine.2023014

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Abstract

Given a bounded open set $ \Omega\subseteq{\mathbb{R}}^n $, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $ \Omega $. We prove that the second eigenvalue $ \lambda_2(\Omega) $ is always strictly larger than the first eigenvalue $ \lambda_1(B) $ of a ball $ B $ with volume half of that of $ \Omega $. This bound is proven to be sharp, by comparing to the limit case in which $ \Omega $ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.

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