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

Abstract

Let $\Omega \subset {\mathbb{R}}^d$ with $d\ge 2$ be a bounded domain of class ${\mathcal{C}}^{1,\beta }$ for some $\beta \in (0,1)$. For $p\in (1,\infty )$ and $s\in (0,1)$, let ${\Lambda }_p^s(\Omega )$ be the first eigenvalue of the mixed local-nonlocal operator $-{\Delta }_p+(-{\Delta }_p{)}^s$ in $\Omega$ 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 annular domains and characterize the rigidity property of the balls in the classical Faber-Krahn inequality for $-{\Delta }_p+(-{\Delta }_p{)}^s$.

13 pages. This article significantly expands upon the concepts presented in our previous paper, arXiv:2305.16672, to address a broader set of cases

References in corpus (1)

• Strict monotonicity of the first $q$-eigenvalue of the fractional $p$-Laplace operator over annuli