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

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.