麦克斯韦系统顶点角处的唯一延拓性

数学理论与应用 ›› 2021, Vol. 41 ›› Issue (3): 38-58.

麦克斯韦系统顶点角处的唯一延拓性

刁怀安¹，刘宏宇^2,*，张龙³

1.吉林大学数学学院, 长春, 130012; 2.香港城市大学数学系, 香港, 999077; 3.辽宁省丹东市凤城一中, 丹东, 118100.

出版日期:2021-09-30 发布日期:2021-10-28

Unique Continuation from a Vertex Corner for Maxwell's System

1. School of Mathematics, Jilin University, Changchun, Jilin 130012, China; 2. Department of Mathematics, City University of Hong Kong, Hong Kong SAR 999077, China; 3.Fengcheng No.1 Middle School, 31 Zhenxing St, Fengcheng, Dandong 118100, China.

Online:2021-09-30 Published:2021-10-28
Contact: Liu Hongyu(1977-), Professor, PhD；E-mail: hongyliu@cityu.edu.hk
Supported by:
The work of Liu Hongyu is supported by Hong Kong RGC General Research Funds (project numbers, 11300821, 12301218 and 12302919)

摘要/Abstract

摘要： 在本文中,我们建立麦克斯韦系统在顶点角处的一类新的唯一延拓性质.当电场或磁场在形成顶点角的平面上满足某种齐次边界条件时,它们在顶点角处一定具有某个消失阶.我们将求得消失阶并建立消失阶和顶点角的多面体角之间的关系.这些结果拓展了[21]中关于边角处的新近结果.这种类型的唯一延拓性质的研究来源于长期存在的电磁波反散射问题.

关键词: 麦克斯韦系统, 时谐电磁散射, 唯一延拓, 消失阶, 顶点角, 反散射, 单次远场测量

Abstract: In this paper, we establish a novel unique continuation property (UCP) for the Maxwell system locally around a vertex corner. It is shown that if certain homogeneous conditions are imposed on the planes forming the vertex corner, the electric/magnetic field must vanish to a certain order. We derive the vanishing order and relate it to the polyhedral angles of the vertex corner. This extends the recent results in [21] where the UCP was considered associated with an edge corner. This type of UCP study originated from the study of a longstanding problem in inverse scattering theory.

Key words: Maxwell system, Time-harmonic electromagnetic scattering, Unique continuation, Vanishing order, Vertex corner, Inverse scattering, A single far-field measurement

刁怀安, 刘宏宇, 张龙 . 麦克斯韦系统顶点角处的唯一延拓性[J]. 数学理论与应用, 2021, 41(3): 38-58.