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The geometric average of curl-free fields in periodic geometries

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Published/Copyright: May 18, 2021
Analysis
From the journal Analysis Volume 41 Issue 3

Abstract

In periodic homogenization problems, one considers a sequence (uη)η of solutions to periodic problems and derives a homogenized equation for an effective quantity u^. In many applications, u^ is the weak limit of (uη)η, but in some applications u^ must be defined differently. In the homogenization of Maxwell’s equations in periodic media, the effective magnetic field is given by the geometric average of the two-scale limit. The notion of a geometric average has been introduced in [G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris 347 2009, 9–10, 571–576]; it associates to a curl-free field Y∖Σ¯→ℝ3, where Y is the periodicity cell and Σ an inclusion, a vector in ℝ3. In this article, we extend previous definitions to more general inclusions, in particular inclusions that are not compactly supported in the periodicity cell. The physical relevance of the geometric average is demonstrated by various results, e.g., a continuity property of limits of tangential traces.

Keywords: Periodic homogenization; Maxwell’s equations; line averages
MSC 2010: 35B27; 78M40; 35B34

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: SCHW 639/6-1

Funding statement: This work was supported by the Deutsche Forschungsgemeinschaft (DFG) in the project “Wellenausbreitung in periodischen Strukturen und Mechanismen negativer Brechung”, grant SCHW 639/6-1.

A Proof of Lemma 2.3

Proof.

We make use of the inclusion ı:U→ℝ3 and the orthogonal projection ı*:ℝ3→U. In particular, ı*⁢ı:U→U is the identity. By our assumption on B:U→ℝ3, the linear map ı*⁢B:U→U is skew-symmetric. We define

B~:=B⁢ı*-ı⁢B*-ı⁢ı*⁢B⁢ı*:ℝ3→ℝ3,

and claim that B~ is skew-symmetric. Indeed, for any x∈ℝ3,

B~⁢x⋅x=B⁢ı*⁢x⋅x-ı⁢B*⁢x⋅x-ı⁢ı*⁢B⁢ı*⁢x⋅x=ı*⁢x⋅B*⁢x-B*⁢x⋅ı*⁢x-ı*⁢B⁢ı*⁢x⋅ı*⁢x=0,

where the last term vanishes by the skew-symmetry of ı*⁢B. We claim that, for x∈U, there holds B~⁢ı⁢x=B⁢x. Indeed, for any y∈ℝ3,

B~⁢ı⁢x⋅y=B⁢x⋅y-ı⁢B*⁢ı⁢x⋅y-ı⁢ı*⁢B⁢x⋅y=B⁢x⋅y-x⋅(ı*⁢B)⁢ı*⁢y-(ı*⁢B)⁢x⋅ı*⁢y=B⁢x⋅y,

since ı*⁢B is skew-symmetric. Because B~:ℝ3→ℝ3 is skew-symmetric, we find a (unique) vector b∈ℝ3 such that B~⁢x=b∧x for every x∈ℝ3. In particular, B⁢x=b∧x for every x∈U.

The set of all vectors b∈ℝ3 that represent B is an affine subspace of ℝ3, it is of the form b0+U∧ with the linear space

U∧={b∈ℝ3:b∧x=0⁢ for every ⁢x∈U}={ℝ3if ⁢dim⁡U=0,Uif ⁢dim⁡U=1,{0}if ⁢dim⁡U≥2.

In the affine space b0+U∧, there is a unique vector b that is orthogonal to U∧. The orthogonal complement of U∧ is (U∧)⊥=U∧⊥, which proves the lemma. ∎

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Received: 2020-11-17
Accepted: 2021-04-25
Published Online: 2021-05-18
Published in Print: 2021-08-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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  2. Combined mixed finite element and nonconforming finite volume methods for flow and transport in porous media
  3. Generalizations and applications of Srinivasa Ramanujan’s integral associated with infinite Fourier sine transforms in terms of Meijer’s G-function
  4. On the integral transform of Mittag-Leffler-type functions with applications
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  7. The geometric average of curl-free fields in periodic geometries
https://doi.org/10.1515/anly-2020-0053
Keywords for this article
Periodic homogenization; Maxwell’s equations; line averages

Articles in the same Issue

  1. Frontmatter
  2. Combined mixed finite element and nonconforming finite volume methods for flow and transport in porous media
  3. Generalizations and applications of Srinivasa Ramanujan’s integral associated with infinite Fourier sine transforms in terms of Meijer’s G-function
  4. On the integral transform of Mittag-Leffler-type functions with applications
  5. Minkowski–Clarkson’s type inequalities
  6. Self-improving properties of discrete Muckenhoupt weights
  7. The geometric average of curl-free fields in periodic geometries
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