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Maxwell's equations with hypersingularities at a negative index material conical tip

Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Mahran Rihani

TL;DR

The paper addresses time-harmonic Maxwell equations across a positive/negative index interface with a conical tip, where critical contrasts yield hypersingular, infinite-energy singularities at the tip and invalidate the classical $L^2$ formulation. By combining $T$-coercivity with Kondratiev theory of detached asymptotics, the authors construct new weighted frameworks that recover well-posedness and align with the limiting absorption principle. They develop a comprehensive scalar-hosted analysis, define out-frameworks for the electric and magnetic problems, prove isomorphism and compactness properties for the principal and compact parts, and establish discrete spectra governing non-invertibility. The results show that standard Maxwell theory is insufficient in this setting, while the proposed frameworks provide robust well-posedness and LAP-compatible solutions, with potential implications for numerical methods and metamaterial design.

Abstract

We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity $\varepsilon$ and the permeability $μ$ take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in $\varepsilon$ and in $μ$, the corresponding scalar operators are not of Fredholm type in the usual $H^1$ spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical $L^2$ framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the $\mathrm{T}$-coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields.

Maxwell's equations with hypersingularities at a negative index material conical tip

TL;DR

The paper addresses time-harmonic Maxwell equations across a positive/negative index interface with a conical tip, where critical contrasts yield hypersingular, infinite-energy singularities at the tip and invalidate the classical formulation. By combining -coercivity with Kondratiev theory of detached asymptotics, the authors construct new weighted frameworks that recover well-posedness and align with the limiting absorption principle. They develop a comprehensive scalar-hosted analysis, define out-frameworks for the electric and magnetic problems, prove isomorphism and compactness properties for the principal and compact parts, and establish discrete spectra governing non-invertibility. The results show that standard Maxwell theory is insufficient in this setting, while the proposed frameworks provide robust well-posedness and LAP-compatible solutions, with potential implications for numerical methods and metamaterial design.

Abstract

We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity and the permeability take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in and in , the corresponding scalar operators are not of Fredholm type in the usual spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the -coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields.
Paper Structure (33 sections, 41 theorems, 199 equations, 4 figures)

This paper contains 33 sections, 41 theorems, 199 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. General notation
  4. Description of the conical tip
  5. Study of the scalar problems
  6. Kondratiev spaces
  7. Scalar operators in Kondratiev spaces
  8. Hypersingularities
  9. Additional properties for the scalar operators in Kondratiev spaces
  10. Mandelstam radiation principle
  11. A new functional framework for the scalar problems
  12. A new framework for the electric problem
  13. Definition of the electric problem
  14. Equivalent formulation of the electric problem
  15. Equivalent norms in TEXT
  16. ...and 18 more sections

Key Result

Proposition 2.1

The embeddings of $\boldsymbol{\mathrm{X}}_N(1)$ in $\boldsymbol{\mathrm{L}}^2(\Omega)$ and of $\boldsymbol{\mathrm{X}}_T(1)$ in $\boldsymbol{\mathrm{L}}^2(\Omega)$ are compact. Moreover there is a constant $C>0$ such that Therefore, in $\boldsymbol{\mathrm{X}}_N(1)$ and in $\boldsymbol{\mathrm{X}}_T(1)$, $\|\boldsymbol{\mathrm{curl}}\,\cdot\|_{\Omega}$ is a norm which is equivalent to $\|\cdot\|

Figures (4)

  • Figure 1: Left: real part of the radial behaviour of the hyper-singularity in 2D ($\Re e\,r^{i\eta}=\cos(\eta\ln r)$). Right: schematic correspondence between the corner and waveguide problems.
  • Figure 2: Internal conical tip (left), conical tips on the boundary (center and right).
  • Figure 3: Schematic picture of the eigenvalues of $\mathscr{L}_\varepsilon$ in the complex plane when Assumption \ref{['AssumptionCritique']} is satisfied. One has something similar for $\mathscr{L}_\mu$.
  • Figure 4: Restriction to $\mathcal{K}_-$ of the imaginary part of a propagating singularity for an internal circular conical tip.

Theorems & Definitions (73)

  • Proposition 2.1
  • Remark 2.2
  • Lemma 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Remark 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Lemma 3.8
  • ...and 63 more