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

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