On the s-derivative of weak solutions of the Poisson problem for the regional fractional Laplacian

2021/12/17 by Temgoua, Remi Yvant
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.09547

Abstract

In this paper, we analyze the s-dependence of the solution u_s to the fractional Poisson equation (-Δ)^s_Ω = f in an open bounded set Ω⊂ ℝ^N. Precisely, we show that the solution map (0,1)→ L²(Ω), s↦ u_s is continuously differentiable. Moreover, when f = λ_s u_s, we also analyze the one-sided differentiability of the first nontrivial eigenvalue of (-Δ)^s_Ω regarded as a function of s ∈ (0,1).

Related

• Differentiability of the nonlocal-to-local transition in fractional Poisson problems
• A new look at the fractional Poisson problem via the Logarithmic\n Laplacian
• Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability
• The Spatially Variant Fractional Laplacian
• Gradient estimates for the fractional p-Poisson equation