Strict Faber–Krahn-type inequality for the mixed local–nonlocal operator under polarization

Abstract Let $\Omega \subset \mathbb{R}^d$ with $d\geq 2$ be a bounded domain of class ${\mathcal C}^{1,\beta }$ for some $\beta \in (0,1)$ . For $p\in (1, \infty )$ and $s\in (0,1)$ , let $\Lambda ^s_{p}(\Omega )$ be the first eigenvalue of the mixed local–nonlocal operator $-\Delta _p+(-\Delta _p)^s$ in Ω with the homogeneous nonlocal Dirichlet boundary condition. We establish a strict Faber–Krahn-type inequality for $\Lambda _{p}^s(\cdot )$ under polarization. As an application of this strict inequality, we obtain the strict monotonicity of $\Lambda _{p}^s(\cdot )$ over the annular domains and characterize the rigidity property of the balls in the classical Faber–Krahn inequality for $-\Delta _p+(-\Delta _p)^s$ .