A Hong-Krahn-Szeg\"{o} inequality for mixed local and nonlocal operators

Given a bounded open set $\Omega\subseteq{\mathbb{R}}^n$, we consider the eigenvalue problem of 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.