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Strong L2 convergence in periodic Dirichlet homogenization

Statement

Under the hypotheses and notation of Periodic scalar Dirichlet homogenization on the unit cube, $u_{ϵ} \to u_{0}$ strongly in $L^{2} (Q)$ .

Facts & Assumptions

Given: The periodic weak solutions satisfy the oscillatory equation, every weak cluster limit is $u_{0}$ , and the cube embedding $H_{0}^{1} (Q) ↪ L^{2} (Q)$ is compact.

[F1]

Periodic scalar Dirichlet homogenization on the unit cube: $u_{ϵ} ⇀ u_{0}$ in $H_{0}^{1} (Q)$ and hence weakly in $L^{2} (Q)$ .

[F2]

Separable H1-zero space and compact L2 inclusion on a cube: every bounded $H_{0}^{1} (Q)$ sequence has a strongly $L^{2}$ -convergent subsequence.

[F3]

Uniformly elliptic periodic scalar diffusion data and Separable H1-zero space and compact L2 inclusion on a cube: testing the oscillatory equation with $u_{ϵ}$ and using $λ \leq a (x / ϵ)$ a.e. and $∥ u_{ϵ} ∥_{2} \leq ∥ \nabla u_{ϵ} ∥_{2}$ gives $∥ \nabla u_{ϵ} ∥_{2} \leq λ^{- 1} ∥ f ∥_{2}$ (with the zero-gradient case immediate).

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): subsequence selections use Countable Choice.

Proof

1.1F1F2F3A1algebra

Suppose strong convergence failed. Then some $δ > 0$ and sequence $ϵ_{n} \to 0$ satisfy $∥ u_{ϵ_{n}} - u_{0} ∥_{2} \geq δ$ for every $n$ . Testing the oscillatory weak equation with $u_{ϵ_{n}}$ and using [F3] gives a uniform $H_{0}^{1} (Q)$ bound: if $∥ \nabla u_{ϵ_{n}} ∥_{2} > 0$ , divide $λ ∥ \nabla u_{ϵ_{n}} ∥_{2}^{2} \leq ∥ f ∥_{2} ∥ u_{ϵ_{n}} ∥_{2} \leq ∥ f ∥_{2} ∥ \nabla u_{ϵ_{n}} ∥_{2}$ by that norm, and if it is zero the bound is immediate. By [F2] and [A1], a subsequence converges strongly in $L^{2}$ to some $v$ . Its weak $L^{2}$ limit is also $u_{0}$ by [F1], so uniqueness of weak limits in the Hilbert space $L^{2} (Q)$ gives $v = u_{0}$ . This contradicts the lower bound $δ$ .

2.1step 1.1algebra∎

The contradiction proves $∥ u_{ϵ} - u_{0} ∥_{2} \to 0$ as $ϵ ↓ 0$ .

Depends on

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Sources

Grégoire Allaire, Introduction to Homogenization Theory, Lecture 1 (standard reference, not scraped)
Zhongwei Shen, Lectures on Periodic Homogenization of Elliptic Systems (standard reference, not scraped)