*Published Paper*

**Inserted:** 19 jun 2023

**Last Updated:** 21 feb 2024

**Journal:** Journal of Geometric Analysis

**Volume:** 34

**Number:** 96

**Pages:** 21

**Year:** 2024

**Doi:** https://doi.org/10.1007/s12220-023-01539-9

**Abstract:**

Let $B, B'\subset \mathbb{R}^d$ with $d\geq 2$ be two balls such that $B'\subset \subset B$ and the position of $B'$ is varied within $B$. For $p\in (1, \infty ),$ $s\in (0,1)$, and $q \in [1, p^*_s)$ with $p^*_s=\frac{dp}{d-sp}$ if $sp < d$ and $p^*_s=\infty $ if $sp \geq d$, let $\lambda ^s_{p,q}(B\setminus \overline{B'})$ be the first $q$-eigenvalue of the fractional $p$-Laplace operator $(-\Delta _p)^s$ in $B\setminus \overline{B'}$ with the homogeneous nonlocal Dirichlet boundary conditions. We prove that $\lambda ^s_{p,q}(B\setminus \overline{B'})$ strictly decreases as the inner ball $B'$ moves towards the outer boundary $\partial B$. To obtain this strict monotonicity, we establish a strict Faber-Krahn type inequality for $\lambda _{p,q}^s(\cdot )$ under polarization. This extends some monotonicity results obtained by Djitte-Fall-Weth (Calc. Var. Partial Differential Equations, 60:231, 2021) in the case of $(-\Delta )^s$ and $q=1, 2$ to $(-\Delta _p)^s$ and $q\in [1, p^*_s).$ Additionally, we provide the strict monotonicity results for the general domains that are difference of Steiner symmetric or foliated Schwarz symmetric sets in $\mathbb{R}^d$.

**Keywords:**
strong comparison principle, Faber-Krahn inequality, Polarizations, nonlocal operator, monotonicity of the first eigenvalue

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