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

Abstract

Let $B,B^{\prime }\subset {\mathbb{R}}^d$ with $d\ge 2$ be two balls such that $B^{\prime }\subset \subset B$ and the position of $B^{\prime }$ is varied within $B$. For $p\in (1,\infty ),$ $s\in (0,1)$, and $q\in [1,p_s^{\ast })$ with $p_s^{\ast }=\frac{dp}{d-sp}$ if $sp<d$ and $p_s^{\ast }=\infty$ if $sp\ge d$, let ${\lambda }_{p,q}^s(B\setminus \overline{B^{\prime }})$ be the first $q$-eigenvalue of the fractional $p$-Laplace operator $(-{\Delta }_p{)}^s$ in $B\setminus \overline{B^{\prime }}$ with the homogeneous nonlocal Dirichlet boundary conditions. We prove that ${\lambda }_{p,q}^s(B\setminus \overline{B^{\prime }})$ strictly decreases as the inner ball $B^{\prime }$ 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 $q\in [1,p_s^{\ast }).$ Additionally, we provide the strict monotonicity results for the general domains that are difference of Steiner symmetric or foliated Schwarz symmetric sets in ${\mathbb{R}}^d$.

14 pages, 2 figures

References in corpus (4)

• A shape variation result via the geometry of eigenfunctions
• On the optimization of the first weighted eigenvalue
• Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality
• On generalized eigenvalue problems of fractional $(p,q)$-Laplace operator with two parameters

Cited by in corpus (1)

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