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

被引:32
作者
Biagi, Stefano [1 ]
Dipierro, Serena [2 ]
Valdinoci, Enrico [2 ]
Vecchi, Eugenio [3 ]
机构
[1] Politecn Milan, Dipartimento Matemat, Via Bonardi 9, I-20133 Milan, Italy
[2] Univ Western Australia, Dept Math & Stat, 35 Stirling Highway, Crawley, WA 6009, Australia
[3] Univ Bologna, Dipartimento Matemat, Piazza Porta San Donato 5, I-40126 Bologna, Italy
来源
MATHEMATICS IN ENGINEERING | 2023年 / 5卷 / 01期
基金
澳大利亚研究理事会;
关键词
operators of mixed order; first eigenvalue; shape optimization; isoperimetric inequality; Faber-Krahn inequality; quantitative results; stability; EQUATIONS;
D O I
10.3934/mine.2023014
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Given a bounded open set Omega subset of 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.
引用
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页数:25
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