In:
Mathematics in Engineering, American Institute of Mathematical Sciences (AIMS), Vol. 5, No. 1 ( 2022), p. 1-25
Abstract:
〈abstract〉〈p〉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.〈/p〉〈/abstract〉
Type of Medium:
Online Resource
ISSN:
2640-3501
DOI:
10.3934/mine.2023014
Language:
Unknown
Publisher:
American Institute of Mathematical Sciences (AIMS)
Publication Date:
2022
detail.hit.zdb_id:
2985862-8
