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

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.