Maxwell's equations with hypersingularities at a conical plasmonic tip

TL;DR

This paper analyzes Maxwell's equations near a conical plasmonic tip with negative dielectric constants, addressing infinite energy singularities by developing a new functional framework using weighted Sobolev spaces and the limiting absorption principle.

Contribution

It introduces a novel functional setting for 3D Maxwell's equations with critical dielectric permittivities, extending scalar case techniques to vector fields with hypersingularities.

Findings

Established a well-posed functional framework for Maxwell's equations with hypersingularities.

Developed new scalar and vector potential representations for singular fields.

Applied the limiting absorption principle to select outgoing solutions.

Abstract

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.