Homogenization of partial differential equations. Translated from the original Russian by M. Goncharenko and D. Shepelsky
This book is devoted to present many not classical homogenization problems, i.e. problems in which coefficients are supposed to satisfy a strict periodicity and the standard boundedness assumptions. Usually these problems depend on a small parameter and the interest is to characterize, if this exists, the limit problem when this parameter goes to zero (this means to obtain some pointwise convergence for the Green's operators). For elliptic or parabolic equations like (\(b_{\varepsilon}\) respectively null or positive) \[ b_{\varepsilon}{{\partial u^{\varepsilon}}\over{\partial t}} - \sum_{i,j=1}^n {{\partial}\over{\partial x_i}} \Big(a_{ij,\varepsilon}{{\partial u^{\varepsilon}}\over{\partial x_j}}\Big) = f, \qquad x\in \Omega_{\varepsilon}, \;t > 0 \] as most of the equations considered in the present book are, the classical results often, but not always, lead to equations which have the same form as the original. Differing from the classical ones, the problems considered in the present book may depend on two (or more) small parameters and the structures may be not strictly periodic or quasi periodic, and the standard boundedness conditions may be violated. In many examples considered media have different characteristics, i.e. these parameters have different order of smallness. This may lead to have limit problem which are not of the same form of the original ones. To make some examples, the authors analyse situations in which the quadratic form associated to \((a_{ij,\varepsilon})_{i,j=1}^n\) is not bounded from above, situations in which the quadratic form associated to \((a_{ij,\varepsilon})_{i,j=1}^n\) is not bounded from below by a positive constant but may go to zero, situations in the nonstationary case in which \(b_{\varepsilon}\) are not bounded from above. In some cases these situations may lead to a nonlocal model, to a multicomponent model, i.e. to a system of two or more equations, or also to a model with memory. It is to be stressed that in general situations like those described above (unbounded coefficients) are very difficult to be attacked, so the authors considered some particular cases and among these only some special situations lead to a non-standard model. Also several examples of homogenization in perforated domains are considered, domains with fine grained (disjont) inclusions (treated also in the random case), connected inclusions (e.g. fibers), and also different types of connectivity are introduced to treat many examples. Also for this class of problems, depending on the situation considered, one can obtain standard or non standard homogenization results. In the last section authors consider operators of order higher than two (again linear) and study problems defined in a domain \(\Omega_s = \Omega \setminus F_s\), \(s \in {\mathbb N}\), where \(F_s\) is a set converging in some sense, when \(s \to + \infty\), to a hypersurface \(\Gamma\) of codimension \(1\) in \(\Omega\).
scientific article; zbMATH DE number 5015850