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

Stefano Biagi, Serena Dipierro, Enrico Valdinoci, Eugenio Vecchi

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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. We prove that the second eigenvalue λ2() 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.

Original languageEnglish
JournalMathematics In Engineering
Volume5
Issue number1
DOIs
Publication statusPublished - 2022

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