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User:Captn wedge/sandbox/Two-scale convergence

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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

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On a non-void, open domain , consider for fixed parameters the boundary value problem

and .

Assume that for all the map has positive definite, symmetric matrices as values. Moreover, we ask that the resulting matrices be essentially bounded and uniformly elliptic independently of , i.e.

, andso the coefficients are uniformly bounded almost everywhere.
.

By the Lax-Milgram lemma, for every the

Historical Definition

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


References

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  • 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.
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