The logarithmic Schrodinger operator and associated Dirichlet problems
The logarithmic Schrodinger operator and associated Dirichlet problems
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DOI：
10.1016/j.jmaa.2022.126656
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发表时间：
2022-09-08
影响因子：
1.3
通讯作者：
Feulefack, Pierre Aime
中科院分区：
文献类型：
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作者：
Feulefack, Pierre Aime
本文研究了与对数符号log（1 + 1）对应的积分微分算子（I-delta）（log）|习|（2）），它是由（I - delta）（logu）（x）= d（N）integral R-Nu（x）- u（x + y）/|y|（2）A（|y|其中d（N）= π- N/2，ω（r）= 2（1- N/2）r（N/2）K（N/2）（r），K-nu是指数为nu的第二类修正贝塞尔函数。这个运算符是方差的Levy生成器？？伽马过程，并作为导数偏微分s？s=0（I - delta）s的分数阶相对论性薛定谔算子在s = 0时.为了研究有界区域上的Dirichlet问题，我们首先引入了泛函分析框架和与（I -δ）log相关的一些性质，这些性质使得我们可以刻画诱导特征值问题和Faber-Krahn型不等式.我们还得到了Poisson问题在RN中的衰减估计，并研究了有界开Lipschitz集上（I - delta）（s）的Dirichlet特征值和特征函数的小阶渐近性s -> 0+. (C)2022爱思唯尔公司All rights reserved.
In this note, we study the integrodifferential operator (I - delta)(log) corresponding to the logarithmic symbol log(1 + |xi|(2)), which is a singular integral operator given by (I - delta)(logu)(x) = d(N) integral R-N u(x) - u(x + y)/|y|(N) omega(|y|) dy, where d(N) = pi- N/2 , omega (r) = 2(1- N/2) r (N/2) K (N/2 )(r) and K-nu is the modified Bessel function of second kind with index nu. This operator is the Levy generator of the variance ?? gamma process and arises as derivative partial differential s?s=0(I - delta)s of fractional relativistic Schrodinger 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 - delta)log, which allow to characterize the induced eigenvalue problem and Faber-Krahn type inequality. We also derive a decay estimate in RN of the Poisson problem and investigate small order asymptotics s -> 0+ of Dirichlet eigenvalues and eigenfunctions of (I - delta)(s) in a bounded open Lipschitz set. (C) 2022 Elsevier Inc. All rights reserved.