The geometric average of curl-free fields in periodic geometries

In periodic homogenization problems, one considers a sequence ( u η ) η {(u^{\eta})_{\eta}} of solutions to periodic problems and derives a homogenized equation for an effective quantity u ^ {\hat{u}} . In many applications, u ^ {\hat{u}} is the weak limit of ( u η ) η {(u^{\eta})_{\eta}} , but in some applications u ^ {\hat{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 ¯ → ℝ 3 {Y\setminus\overline{\Sigma}\to\mathbb{R}^{3}} , where Y is the periodicity cell and Σ an inclusion, a vector in ℝ 3 {\mathbb{R}^{3}} . 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.