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

<abstract><p>Given a bounded open set $ \Omega\subseteq{\mathbb{R}}^n $, we consider the eigenvalue problem for nonlinear mixed local/nonlocal operator with vanishing conditions in complement of \Omega $. We prove that second \lambda_2(\Omega) is always strictly larger than first \lambda_1(B) ball B volume half This bound proven to be sharp, by comparing limit case which consists two equal balls far from each other. More precisely, differently local case, an optimal shape does not exist, but minimizing sequence given union disjoint whose mutual distance tends infinity.</p></abstract>