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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]

The problem of periodic homogenisation for a conducting two-phase composite exhibiting non-locality in the overall behaviour is considered. The matrix is isotropic and its conductivity is constant, while the fibre-shaped inclusions are assumed to be anisotropic, with high contrast between conductivities along the fibres and in the transverse directions. The contrast is set to correspond to the so-called double-porosity scaling, which is of special interest. A full two-scale asymptotic expansion for the solution is constructed and rigorously justified via appropriate error estimates. An example of the non-local homogenised operator in the case of fibres with circular cross-sections is given. [ABSTRACT FROM AUTHOR]

We consider a scalar quasilinear equation in the divergence form with periodic rapid oscillations, which may be a model of, e.g., nonlinear conducting, dielectric, or deforming in a restricted way hardening elastic-plastic composites, with “outer” periodicity conditions of a fixed large period. Under some natural growth assumptions on the stored-energy function, we construct for uniformly elliptic problems a full two-scale asymptotic expansion, which has a precise “double-series” structure, separating the slow and the fast variables “in all orders”, so that its “slowly varying” part solves asymptotically an “infinite-order homogenised equation” (cf. Bakhvalov, N.S., Panasenko, G.P.:Homogenisation: Averaging Processes in Periodic Media. Nauka, Moscow, 1984 (in Russian); English translation: Kluwer, 1989), and whose higher-order terms depend on the higher gradients of the slowly varying part. We prove the error bound, i.e., that the truncated asymptotic expansion is “higher-order” close to the actual solution in appropriate norms. The approach is extended to a non-uniformly elliptic case: for two-dimensional power-law potentials we prove the “non-degeneracy” using topological index methods. Examples and explicit formulae for the higher-order terms are given. In particular, we prove that the first term in the higher-order homogenised equations is related to the first-order corrector to the “mean” flux, and has in general the form of a fully nonlinear operator which is quadratic with respect to its highest (second) derivative being a linear combination of the second minors of the Hessian with coefficients depending on the first gradient, and in dimension two is of Monge-Ampère type. We show that this term is present at least for some examples (three-phase power-law laminates).In the second part of the paper we extend to this nonlinear context some of the results previously developed by us in the linear case (Smyshlyaev, V.P., Cherednichenko, K.D.J. Mech. Phys. Solids48, 1325-1357, 2000). In particular, we prove that the slowly varying part of the full asymptotic expansion is the rigorous asymptotics “in all orders” for the “translationally averaged” actual solution and flux, or in the sense of a higher-order version of the weak convergence. We then explore to what extent the method of variational truncation of the infinite-order homogenised equation, successfully implemented by us in the linear context in the previous work for constructing explicit “higher-order” homogenised equations, is extendable to the nonlinear regime. We propose a natural extension and prove that at least under some further natural “non-degeneracy” assumptions it has a solution (the existence), and that any such solution is close to the actual solution in appropriate norms. [ABSTRACT FROM AUTHOR]