Open AccessA Hong-Krahn-Szegö inequality for mixed local and nonlocal operators
Author(s) -
Stefano Biagi,
Serena Dipierro,
Enrico Valdinoci,
Eugenio Vecchi
Publication year - 2022
Publication title -
mathematics in engineering
Language(s) - English
Resource type - Journals
ISSN - 2640-3501
DOI - 10.3934/mine.2023014
Subject(s) - mathematics , omega , eigenvalues and eigenvectors , ball (mathematics) , bounded function , disjoint sets , combinatorics , operator (biology) , lambda , infinity , limit (mathematics) , mathematical analysis , pure mathematics , physics , quantum mechanics , biochemistry , chemistry , repressor , transcription factor , gene
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.