Some remarks on two-scale convergence and periodic unfolding

On two-scale convergence and periodic unfolding

“…The Definition 4.5 thus yields a direct proof of principal results of the two-scale convergence theory, see [9]: …”

“…In [7], [8] and [9] a modified extension of the unfolding operator T ε is proposed which removes the problems cited above. In the uncomplete cells Λ ε the unfolding is extended by the identity:…”

“…Due to Theorem 6.1 {∇u ε } contains a subsequence weakly two-scale converging to a function η. Since the coefficients are strongly two-scale converging, Theorem 6.2 enables to pass to the limit in (1), see [9].…”

“…Thanks to properties of the unfolding T * ε cited in Lemma 4.4, the proofs of Theorems 6.1 and 6.2 easily follow from the theory of L p -spaces. The unfolding operator T * ε and averaging operator U * ε satisfying measure and integral conservation property, see Lemma 4.4 and Lemma 5.1, are starting point for general convergence for nonperiodic homogenization, see [9].…”

Homogenization on perforated domains

“…The operator was improved by J. Franc· and N.Svanstedt in [FS12]. This change simplies the proofs and removes several diculties and necessity of introducing unfolding criterion for integrals" (see e.g.…”