DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Two-scale weak convergence for periodic tests

Definition

Let $Q = Y = (0, 1)^{d}$ , let $ϵ_{n} > 0$ tend to zero, and let $(u_{n})$ be bounded in $L^{2} (Q)$ . We say $u_{n}$ two-scale converges weakly to $u \in L^{2} (Q \times Y)$ along $(ϵ_{n})$ if for every finite sum $Φ (x, y) = \sum_{r = 1}^{m} ϕ_{r} (x) ψ_{r} (y), ϕ_{r} \in C_{c}^{\infty} (Q), ψ_{r} \in C_{#}^{\infty} (Y),$ one has $\lim_{n \to \infty} \int_{Q} u_{n} (x) Φ (x, x / ϵ_{n}) d x = \int_{Q} \int_{Y} u (x, y) Φ (x, y) d y d x .$ Here $C_{#}^{\infty} (Y)$ is the smooth $Z^{d}$ -periodic class from Zero-mean periodic Sobolev cell space. The test definition is independent of how a finite sum is represented because both sides are linear in $Φ$ .

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)