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Using a suitable stochastic version of the compactness argument of [Zhikov VV. On an extension of the method of two-scale convergence and its applications. Sb Math. 2000;191(7-8):973-1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled 'macroscopic' and 'microscopic' components of the field, giving rise to an analogue of the 'Zhikov function', which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit. [ABSTRACT FROM AUTHOR]

Following a number of recent studies of resolvent and spectral convergence of nonuniformly elliptic families of differential operators describing the behavior of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a "defect"": an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period ε goes to zero, has a band-gap structure. We consider a sequence of eigenvalues λε that are induced by the defect and converge to a point λ0 located in a gap of the limit spectrum for the periodic case. We show that the corresponding eigenfunctions are "extremely"" localized to the defect, in the sense that the localization exponent behaves as exp(-v/ε), v > 0, which has not been observed in the existing literature. In two- and three-dimensional configurations, whose one-dimensional cross sections are described by the setting considered, this implies the existence of propagating waves that are localized to a vicinity of the defect. We also show that the unperturbed operators are norm-resolvent close to a degenerate operator on the real axis, which is described explicitly. [ABSTRACT FROM AUTHOR]

We prove operator-norm resolvent convergence estimates for one-dimensional periodic differential operators with rapidly oscillating coefficients in the non-uniformly elliptic high-contrast setting, which has been out of reach of the existing homogenisation techniques. Our asymptotic analysis is based on a special representation of the resolvent of the operator in terms of the M-matrix of an associated boundary triple ('Krein resolvent formula'). The resulting asymptotic behaviour is shown to be described, up to a unitary transformation, by a non-standard version of the Kronig-Penney model on $${\mathbb{R}}$$ . [ABSTRACT FROM AUTHOR]

For the system of Maxwell equations of electromagnetism in an l-periodic composite medium of overall size L (0 < l < L < ∞), in the low-frequency quasi-static approximation, we develop an electromagnetic version of strain-gradient theories, where the magnetic field is not a function of the magnetic induction alone but also of its spatial gradients, and the electric field depends not only on the displacement but also on displacement gradients. Following the work [V. P. Smyshlyaev and K. D. Cherednichenko, J. Mech. Phys. Solids, 48 (2000), pp. 1325{1357], we propose a combination of variational and asymptotic approaches to the multiscale analysis of the Maxwell system. We provide rigorous convergence estimates of higher order of smallness with respect to the inverse of the "scale separation parameter" L=l: Using a special "ensemble averaging" procedure for a family of periodic problems, we derive an infinite-order version of the standard homogenized operator of second order. [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]