LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)

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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.

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Planar rotation carries H(div) normal traces to tangential traces

Statement

Assume Countable Choice. For a nondegenerate oriented triangle $K \subset R^{2}$ , set $R (a, b) = (- b, a)$ , let $n$ be cell-outward and $t = R n$ . If $σ \in H (div; K)$ is real, then $R σ \in H (curl; K)$ , $curl (R σ) = div σ$ distributionally, and $\int_{K} (R σ \cdot R \nabla ϕ + curl (R σ) ϕ) = ⟨ γ_{n} σ, [ϕ] ⟩$ for all $ϕ \in H^{1} (K)$ . Thus weak tangential trace equals weak normal trace under $t = R n$ .

Here $H (curl; K) : = {w \in L^{2} (K; R^{2}) : \partial_{1} w_{2} - \partial_{2} w_{1} \in L^{2} (K)}$ with its graph norm. The weak tangential trace is the functional on $H^{1} (K) / H_{0}^{1} (K)$ defined by the integral in the Statement; the computation below proves it is the rotation of the PE08 weak normal trace.

Facts & Assumptions

Given: Countable Choice and real $σ \in H (div; K)$ .

[A1]

Countable Choice is declared for trace interfaces. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

H(div) consists of real L2 vectors with L2 distributional divergence. (The real H(div) space)

[F2]

The weak normal trace acts by $\int_{K} (σ \cdot \nabla ϕ + div σ ϕ)$ . (Weak normal trace by Green duality)

Proof

1.1givenF1algebra

Writing $R σ = (- σ_{2}, σ_{1})$ gives $curl (R σ) = \partial_{1} σ_{1} + \partial_{2} σ_{2} = div σ$ as distributions; [F1] places it in $L^{2}$ .

2.1A1F2step 1.1algebra∎

Orthogonality gives $R σ \cdot R \nabla ϕ = σ \cdot \nabla ϕ$ , so step 1.1 makes the trace integral exactly [F2]. Also $R σ \cdot t = R σ \cdot R n = σ \cdot n$ .

Depends on

Used by

Ultraweak Poisson DPG spaces and plan-consistent bilinear form Definition

Dependency tree · two levels

14 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

Jay Gopalakrishnan and Weifeng Qiu, An analysis of the practical DPG method (standard reference, not scraped)
Leszek Demkowicz and Jay Gopalakrishnan, Analysis of the DPG method for the Poisson equation (standard reference, not scraped)
Leszek Demkowicz, Mathematical Theory of Finite Elements (standard reference, not scraped)
C. Carstensen, L. Demkowicz, and J. Gopalakrishnan, A Posteriori Error Control for DPG Methods (standard reference, not scraped)