The logarithmic Schrödinger operator and associated Dirichlet problems

2021/12/16 by Feulefack, Pierre Aime · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.08783

Abstract

In this note, we study the integrodifferential operator (I-Δ)^log corresponding to the logarithmic symbol log(1+|ξ|²), which is a singular integral operator given by (I-Δ)^log u(x)=d_N∫_ℝ^N\fracu(x)-u(x+y)|y|^Nω(|y|) dy, where d_N=π^-(N)/(2), ω(r)=2^1-(N)/(2)r^(N)/(2)K_(N)/(2)(r) and K_ν is the modified Bessel function of second kind with index ν. This operator is the Lévy generator of the variance gamma process and arises as derivative ∂_s|_s=0(I-Δ)^s of fractional relativistic Schrödinger operators at s=0. In order to study associated Dirichlet problems in bounded domains, we first introduce the functional analytic framework and some properties related to (I-Δ)^log, which allow to characterize the induced eigenvalue problem and Faber-Krahn type inequality. We also derive a decay estimate in ℝ^N of the Poisson problem and investigate small order asymptotics s→ 0⁺ of Dirichlet eigenvalues and eigenfunctions of (I-Δ)^s in a bounded open Lipschitz set.

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• Sharp embeddings and existence results for Logarithmic p-Laplacian equations with critical growth

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