An Introduction to Γ-Convergence pp 247-255 | Cite as

# Homogenization

Chapter

## Abstract

In this chapter we study the Γ-limit, as ε→0 where

^{+},of a family (*F*_{ε})_{ε>0}of functional of the form$$ {{F}_{\varepsilon }}(u,A) = \left\{ {\begin{array}{*{20}{c}} {\int_{A} {f(\frac{x}{\varepsilon },Du(x)),ifu \in W_{{loc}}^{{1,1}}(A),} } \hfill \\ { + \infty ,otherwise,} \hfill \\ \end{array} } \right. $$

*f*(*x*,ξ) is convex in ξ and periodic in*x*. When*F*_{ ε }represents the stored energy of a (possibly nonlinear) inhomogeneous material with a periodic structure, this convergence analysis is related to the so called “homogenization problem”, i.e., the problem of finding the physical properties of a homogeneous material, whose overall response is close to that of the periodic material, when the size ε of the periodicity cell tends to 0.## Keywords

Open Subset Positive Real Number Convergence Analysis Lower Semicontinuous Homogeneous Material
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1993