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

TL;DR

This paper proves a strict Faber-Krahn inequality for a mixed local-nonlocal eigenvalue problem under polarization, demonstrating strict monotonicity and characterizing the optimality of balls for certain domains.

Contribution

It establishes a new strict inequality for the first eigenvalue of a mixed operator under polarization, extending classical Faber-Krahn results to nonlocal operators.

Findings

Strict Faber-Krahn inequality under polarization.

Monotonicity of eigenvalues over annular domains.

Characterization of balls as extremal domains.

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 $\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 $-\Delta _p+(-\Delta _p)^s$.