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

Abstract

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