2023/05/26 by K. Kumar, Nirjan Biswas, Kumar, K Ashok +1
Computer Science · Mathematics · #2020: Primary 35R11 #35B06 #47J10 #49Q10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Secondary 35B51
paper · pdf · doi:10.48550/arxiv.2305.16672
openalex publication_date 2023/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let B, B'⊂ ℝd with d≥ 2 be two balls such that B'⊂ ⊂ B and the position of B' is varied within B. For p∈ (1, ∞ ), s∈ (0,1), and q ∈ [1, p^*s) with p^*s=(dp)/(d-sp) if sp < d and p^*s=∞ if sp ≥ d, let λsp,q(B∖ B') be the first q-eigenvalue of the fractional p-Laplace operator (-Δp)s in B∖ B' with the homogeneous nonlocal Dirichlet boundary conditions. We prove that λsp,q(B∖ B') strictly decreases as the inner ball B' moves towards the outer boundary ∂ B. To obtain this strict monotonicity, we establish a strict Faber-Krahn type inequality for λp,qs(⋅ ) under polarization. This extends some monotonicity results obtained by Djitte-Fall-Weth (Calc. Var. Partial Differential Equations, 60:231, 2021) in the case of (-Δ)s and q=1, 2 to (-Δp)s and q∈ [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 ℝd.