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

Let B, B' subset of R-d with d >= 2 be two balls such that B' subset of subset of B and the position of B' is varied within B. For p is an element of (1, infinity), s is an element of (0,1), and q is an element of [1, p(s)*) with p(s)* = dp/d - sp if sp < d and p(s)* = infinity if sp >= d, let lambda(s)(p,q) (B\(B') over bar) be the first q-eigenvalue of the fractional p-Laplace operator (-Delta(p))(s) in B\(B') over bar with the homogeneous nonlocal Dirichlet boundary conditions. We prove that lambda(s)(p,q) (B\(B') over bar) strictly decreases as the inner ball B' moves towards the outer boundary partial derivative B. To obtain this strict monotonicity, we establish a strict Faber-Krahn type inequality for lambda(s)(p,q) (center dot) 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 is an element of [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 R-d.