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

Given a bounded open set $ \Omega\subseteq{\mathbb{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.