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For arbitrarily small values of ε > 0 , we formulate and analyse the Maxwell system of equations of electromagnetism on ε -periodic sets S ε ⊂ R 3. Assuming that a family of Borel measures μ ε , such that supp (μ ε) = S ε , is obtained by ε -contraction of a fixed 1-periodic measure μ , and for right-hand sides f ε ∈ L 2 (R 3 , d μ ε) , we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic "singular structures", when μ is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for L 2 functions, an associated Poincaré-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method. [ABSTRACT FROM AUTHOR]

We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the "stiff" material, and a "soft" material that fills the remaining pores. We assume that the pores are of size 0 < ε << 1 and are periodically distributed with period ε. We also assume that the stiffness of the soft material degenerates with rate ε2γ,γ > 0, so that the contrast between the two materials becomes infinite as ε ↓ 0. We study the homogenisation limit ε ↓ 0 in a low energy regime, where the displacement of the stiff component is infinitesimally small. We derive an effective two-scale model, which, depending on the scaling of the energy, is either a quadratic functional or a partially quadratic functional that still allows for large strains in the soft inclusions. In the latter case, averaging out the small scale-term justifies a single-scale model for high-contrast materials, which features a non-linear and non-monotone effect describing a coupling between microscopic and the effective macroscopic displacements. [ABSTRACT FROM AUTHOR]