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Wavenumber-explicit analytic regularity of the heterogeneous Maxwell equations with impedance boundary conditions

Jens Markus Melenk, David Wörgötter

Abstract

We consider the time-harmonic Maxwell equations at a nonzero wavenumber $k\in\mathbb{C}$ on a bounded and simply connected Lipschitz domain $Ω$ with an analytic boundary $Γ$, on which we impose impedance boundary conditions. We suppose that the (possibly complex-valued) permeability and permittivity tensor fields $\boldsymbolμ^{-1}$ and $\boldsymbol{\varepsilon}$ are piecewise analytic in $Ω$ and discontinuous only across certain mutually disjoint analytic surfaces inside of $Ω$. We show that under these circumstances, any weak solution of Maxwell's equations is piecewise analytic in $Ω$ and that the growth of its derivatives can be controlled explicitly in the wavenumber $k$.

Wavenumber-explicit analytic regularity of the heterogeneous Maxwell equations with impedance boundary conditions

Abstract

We consider the time-harmonic Maxwell equations at a nonzero wavenumber on a bounded and simply connected Lipschitz domain with an analytic boundary , on which we impose impedance boundary conditions. We suppose that the (possibly complex-valued) permeability and permittivity tensor fields and are piecewise analytic in and discontinuous only across certain mutually disjoint analytic surfaces inside of . We show that under these circumstances, any weak solution of Maxwell's equations is piecewise analytic in and that the growth of its derivatives can be controlled explicitly in the wavenumber .
Paper Structure (35 sections, 39 theorems, 295 equations)

This paper contains 35 sections, 39 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. General notation and main results
  3. ${\mathcal{A}}$-partitions and classes of piecewise analytic functions
  4. General assumptions on the coefficients and main results
  5. Surface differential operators and traces of ${\bf H}(\operatorname{curl}, \Omega)$
  6. Trace spaces of ${\bf H}(\operatorname{curl}, \Omega)$
  7. Variational formulation of Maxwell's equations
  8. Analytic regularity in the interior of subdomains
  9. Commutator fields and Sobolev-Morrey seminorms
  10. Regularity of vector fields with piecewise regular curl and divergence
  11. Auxiliary estimates for interior analyticity
  12. Proof of Theorem \ref{['Mainresult1']}
  13. Local flattenings of boundaries and subdomain interfaces
  14. Definition and properties of local flattenings
  15. Transformation invariance of Maxwell's equations
  16. ...and 20 more sections

Key Result

Theorem 2.13

Let $\Omega, \Omega_2\subseteq\mathbb{R}^3$ be bounded Lipschitz domains with $\Omega_2\subset\subset\Omega$ and let $\boldsymbol{\mu}^{-1}, \boldsymbol{\varepsilon}\in{\boldsymbol{\mathcal{A}}}\left(\Omega;\mathbb{C}^{3\times 3}\right)$ satisfy coercivedomain. Under these assumptions, consider a ve for all ${\bf v}\in{\bf H}(\operatorname{curl}, \Omega)$ with compact support in $\Omega$, where th

Theorems & Definitions (66)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 56 more