User:Captn wedge/sandbox/Two-scale convergence

Two-scale convergence is a mathematically rigorous technique for the homogenization of partial differential equations involving periodically oscillating coefficients. Historically, first predecessors of two-scale convergence were developed since the late 1960's and 1970's in different contexts and for different problems. However, its stringent formulation took place around 1990. By 2002, two-scale convergence was also reformulated by periodic unfolding. Both formulations are in use nowadays.

Motivation
On a non-void, open domain $$ \Omega \subset \R ^n $$, consider for fixed parameters $$ \varepsilon >0 $$ the boundary value problem


 * $$ -\nabla \cdot \left[ A _\varepsilon \left( x \right) \nabla u _{\varepsilon} \right] = f(x)  \,$$ and $$ u _{\vert \Omega} \equiv 0 $$.

Assume that for all $$ \varepsilon >0 $$ the map $$ A _\varepsilon : \R ^n \rightarrow \R _{sym,pd} ^{n\times n} $$ has positive definite, symmetric matrices as values. Moreover, we ask that the resulting matrices be essentially bounded and uniformly elliptic independently of $$ \varepsilon >0 $$, i.e.


 * $$ \exists C _1 \geq 0 : \forall i,j \in \lbrace 1,\ldots,n\rbrace : \quad \Vert A_{i,j} \Vert _{L^\infty (\R ^n)} \leq C_1  \qquad$$, andso the coefficients are uniformly bounded almost everywhere.
 * $$ \exists C _2 >0 : \forall \xi \in \R ^n: \quad C_2 \Vert \xi \Vert ^2 \leq (A(y)\xi, \xi ) \qquad\text{ for almost all } y\in \R ^n   $$.

By the Lax-Milgram lemma, for every $$ \varepsilon >0 $$ the

Historical Definition
1. Historical definition given by Allaire 2. Hint at periodic unfolding.