利普希兹区域上薛定谔方程Neumann问题的加权估计

数学物理学报 ›› 2016, Vol. 36 ›› Issue (6): 1165-1185.

利普希兹区域上薛定谔方程Neumann问题的加权估计

黄文礼^1,2, 陶祥兴³

1. 浙江财经大学金融学院杭州 310018;
2. 浙江财经大学中国金融研究院杭州 310018;
3. 浙江科技学院理学院杭州 310023

收稿日期:2016-04-01 修回日期:2016-08-07 出版日期:2016-12-26 发布日期:2016-12-26
通讯作者: 黄文礼 E-mail:matahhuang@zju.edu.cn
作者简介:陶祥兴,xxtau@163.com
基金资助:
浙江省教育厅项目（Y201431706）资助

Weighted Estimates on the Neumann Problem for L² for Schrödinger Equations in Lipschitz Domains

Huang Wenli^1,2, Tao Xiangxing³

1. School of Finance, Zhejiang University of Finance & Economics, Hangzhou 310018;
2. China Academy of Financial Research, Zhejiang University of Finance & Economics, Hangzhou 310018;
3 School of Sciences, Zhejiang University of Sciences & Technology, Hangzhou 310023

Received:2016-04-01 Revised:2016-08-07 Online:2016-12-26 Published:2016-12-26
Supported by:
Supported by the Project of Education Department of Zhejiang Province (Y201431706)

摘要/Abstract

摘要：

该文研究了利普希兹区域上加权空间H^p（∂Ω，ω_αdσ）和L^p（∂Ω，ω_αdσ）（1-ε < p≤2）上薛定谔方程-Δu+Vu=0加权估计问题.记Ω是Rⁿ（n≥3）上边界连通的有界利普希兹区域.令ω_α（Q）=|Q-Q₀|^α，这里Q₀是∂Ω上的一个不动点.对于定义在Ω上的薛定谔方程-Δu+Vu=0，其中奇异非负位势V属于反Hölder类-B_n.该文研究边值落在加权空间H^p（∂Ω，ω_αdσ）或L^p（∂Ω，ω_αdσ）上的Neumann问题，这里dσ表示∂Ω上的测度.对于特定范围的α，方程存在唯一解u，使得非切向的极大函数▽u在H^p（∂Ω，ω_αdσ）或L^p（∂Ω，ω_αdσ）上.此外，还建立了这些解的一致估计.

关键词: Neumann问题, 薛定谔方程, 利普希兹区域, 加权估计

Abstract:

In this paper, we consider the weighted estimates in the weighted space H^p(∂Ω,ω_αdσ) or L^p(∂Ω,ω_αdσ)(1-ε < p≤2) for Schrödinger equation -Δu+Vu=0 on Lipschitz domains. Let Ω be a bounded Lipschitz domain with connected boundary in Rⁿ, n≥3. Let ω_α(Q)=|Q-Q₀|^α, where Q₀ is a fixed point on ∂Ω. For Schrödinger equation -Δu+Vu=0 in Ω, with singular non-negative potentials V belonging to the reverse Hölder class B_n, we study the Neumann problem with boundary date lies in the weighted space H^p(∂Ω,ω_αdσ) or L^p(∂Ω,ω_αdσ), where dσ denotes the surface measure on ∂Ω. We show that for certain ranges of α, there is a unique solution u, such that the non-tangential maximal function of ▽u is in H^p(∂Ω,ω_αdσ) or L^p(∂Ω,ω_αdσ). Moreover, the uniform estimates are founded.

Key words: Neumann Problem, Schrödinger Equation, Lipschitz Domains, Weighted Estimates

中图分类号:

O174.2

黄文礼, 陶祥兴. 利普希兹区域上薛定谔方程Neumann问题的加权估计[J]. 数学物理学报, 2016, 36(6): 1165-1185.

Huang Wenli, Tao Xiangxing. Weighted Estimates on the Neumann Problem for L² for Schrödinger Equations in Lipschitz Domains[J]. Acta mathematica scientia,Series A, 2016, 36(6): 1165-1185.

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