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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.
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- 2 Active
1/01/19 → 19/05/25
30/11/18 → 30/11/22