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Optimal estimate of electromagnetic field concentration between two nearly-touching inclusions in the quasi-static regime

Youjun Deng, Hongyu Liu, Liyan Zhu

Abstract

We investigate the electromagnetic field concentration between two nearly-touching inclusions that possess high-contrast electric permittivities in the quasi-static regime. By using layer potential techniques and asymptotic analysis in the low-frequency regime, we derive low-frequency expansions that provide integral representations for the solutions of the Maxwell equations. For the leading-order term $\bE_0$ of the asymptotic expansion of the electric field, we prove that it has the blow up order of $ε^{-1} |\ln ε|^{-1}$ within the radial geometry, where $ε$ signifies the asymptotic distance between the inclusions. By delicate analysis of the integral operators involved, we further prove the boundedness of the first-order term $\bE_1$. We also conduct extensive numerical experiments which not only corroborate the theoretical findings but also provide more discoveries on the field concentration in the general geometric setup. Our study provides the first treatment in the literature on field concentration between nearly-touching material inclusions for the full Maxwell system.

Optimal estimate of electromagnetic field concentration between two nearly-touching inclusions in the quasi-static regime

Abstract

We investigate the electromagnetic field concentration between two nearly-touching inclusions that possess high-contrast electric permittivities in the quasi-static regime. By using layer potential techniques and asymptotic analysis in the low-frequency regime, we derive low-frequency expansions that provide integral representations for the solutions of the Maxwell equations. For the leading-order term of the asymptotic expansion of the electric field, we prove that it has the blow up order of within the radial geometry, where signifies the asymptotic distance between the inclusions. By delicate analysis of the integral operators involved, we further prove the boundedness of the first-order term . We also conduct extensive numerical experiments which not only corroborate the theoretical findings but also provide more discoveries on the field concentration in the general geometric setup. Our study provides the first treatment in the literature on field concentration between nearly-touching material inclusions for the full Maxwell system.
Paper Structure (21 sections, 17 theorems, 148 equations, 8 figures)

This paper contains 21 sections, 17 theorems, 148 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Mathematical setup
  3. Main result
  4. Boundary integral operators
  5. Definitions
  6. Boundary integral identities
  7. Low-frequency asymptotic expansion
  8. Asymptotics for the operators
  9. Asymptotics for the fields $(\mathbf{E}^{i}, \mathbf{H}^{i})$
  10. The representation of the solution
  11. Asymptotic expansions and representation for the potential $\left[{\phi} , {\psi}\right]^{'}$
  12. Asymptotics for the solution $\mathbf{E}(\mathbf{x})$
  13. Estimate of $\|\mathbf{E}\|_{L^\infty(\Omega\backslash \overline{B_1\cup B_2})}$
  14. Analysis of $\mathbf{E}_0$
  15. The analysis of $\mathbf{E}_1$
  16. ...and 6 more sections

Key Result

Theorem 3.1

Suppose $\omega \ll 1$, and $\varepsilon_c=O(\frac{1}{\omega})=\frac{\tilde{C} }{\omega}$. Let $\mathbf{E}\in H_{loc}\left ( curl;\mathbb{R}^3 \right )$ be a solution of eq:maxwell, for sufficiently small $\epsilon>0$, there exist positive constants $C_1$, $C_2$ depending only on $r$ and $\Omega$ su

Figures (8)

  • Figure 2.1: Model: Two close-to-touching spheres embedded in the background.
  • Figure 7.1: Spherical domains. Contour: Norm of electric field (V/m).
  • Figure 7.2: Spherical domains. The variation of electric field norm with $\epsilon$.
  • Figure 7.3: Hemisphere domains. Contour: Norm of electric field (V/m).
  • Figure 7.4: Hemisphere domains. The variation of electric field norm with $\epsilon$.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Theorem 3.1
  • Remark 3.1
  • Proposition 4.1
  • Proposition 4.2
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • Lemma 6.3
  • proof
  • Lemma 6.4
  • ...and 24 more