论文标题
麦克斯韦方程，具有锥形等离子尖端的超连贯性
Maxwell's equations with hypersingularities at a conical plasmonic tip
论文作者
论文摘要
在这项工作中，我们对在具有负介电常数的材料的圆锥形尖端的情况下对时谐波麦克斯韦的方程进行分析感兴趣。当这些常数属于一定的临界范围时，电磁场在具有无限能量的尖端上表现出强烈的振荡奇异性。因此，麦克斯韦的方程式在经典$ l^2 $框架中没有很好。当前工作的目的是为3D Maxwell的方程提供适当的功能设置，而介电介电常数（但不是磁渗透性）需要关键值。遵循2D标量案例所做的事情，其想法是在加权的Sobolev空间中工作，从而增加了所谓的外向发出的繁殖奇点。该分析需要标量和矢量潜在表示的新结果。外向行为是通过限制吸收原理选择的。
In this work, we are interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip which have infinite energy. Consequently Maxwell's equations are not well-posed in the classical $L^2$ framework. The goal of the present work is to provide an appropriate functional setting for 3D Maxwell's equations when the dielectric permittivity (but not the magnetic permeability) takes critical values. Following what has been done for the 2D scalar case, the idea is to work in weighted Sobolev spaces, adding to the space the so-called outgoing propagating singularities. The analysis requires new results of scalar and vector potential representations of singular fields. The outgoing behaviour is selected via the limiting absorption principle.
