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

Given a bounded open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $\mathrm{\Omega }$. We prove that the second eigenvalue ${\lambda }_2(\mathrm{\Omega })$ is always strictly larger than the first eigenvalue ${\lambda }_1(B)$ of a ball $B$ with volume half of that of $\mathrm{\Omega }$. This bound is proven to be sharp, by comparing to the limit case in which $\mathrm{\Omega }$ 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.