Strict Monotonicity of the First q-Eigenvalue of the Fractional p-Laplace Operator Over Annuli

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