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 $\Omega \subset \mathbb {R} ^{n}$, 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 $A_{\varepsilon }:\mathbb {R} ^{n}\rightarrow \mathbb {R} _{sym,pd}^{n\times n}$ 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.

$\exists C_{1}\geq 0:\forall i,j\in \lbrace 1,\ldots ,n\rbrace :\quad \Vert A_{i,j}\Vert _{L^{\infty }(\mathbb {R} ^{n})}\leq C_{1}\qquad$, andso the coefficients are uniformly bounded almost everywhere.

$\exists C_{2}>0:\forall \xi \in \mathbb {R} ^{n}:\quad C_{2}\Vert \xi \Vert ^{2}\leq (A(y)\xi ,\xi )\qquad {\text{ for almost all }}y\in \mathbb {R} ^{n}$.

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

Historical Definition

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

References

• G. Allaire: Homogenization and two-scale convergence, SIAM J. Math. Anal., Vol. 23, No. 6, pp. 1482-1515, 1992.
• A. Bensoussan, J. L. Lions, G. Papanicolaou: Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
• D. Cioranescu , A. Damlamian , G. Griso: The periodic unfolding method in homogenization, SIAM J. Math. Anal., Vol. 40, No. 4, pp. 1585–1620, 2008.
• A. Mielke, A. Timofte: Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, WIAS preprint No. 1172, Berlin, 2006.
• G. Nguetseng: A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20, pp. 608-623., 1989.

External links

• www.example.com