@article{Biagi2023, 
author = {Stefano Biagi and Serena Dipierro and Enrico Valdinoci and Eugenio Vecchi},
title = {A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {1},
pages = {1-25},
keywords = {operators of mixed order, first eigenvalue, shape optimization, isoperimetric inequality, Faber-Krahn inequality, quantitative results, stability},
url = {https://www.sciopen.com/article/10.3934/mine.2023014},
doi = {10.3934/mine.2023014},
abstract = {Given a bounded open set  Ω⊆Rn, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of  Ω. We prove that the second eigenvalue  λ2(Ω) is always strictly larger than the first eigenvalue  λ1(B) of a ball  B with volume half of that of  Ω. This bound is proven to be sharp, by comparing to the limit case in which  Ω 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.}
}