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Weak curl-free fields on a contractible polyhedron are gradients

Statement

Assume the Axiom of Choice. Let $Ω \subset R^{3}$ be a nonempty bounded connected contractible Lipschitz polyhedron. If $u \in H (curl; Ω)$ has distributional curl zero, then there is $p \in H^{1} (Ω)$ with $u = \nabla p$ almost everywhere; $p$ is unique up to an additive constant.

Facts & Assumptions

Given: AC, the domain and the real curl-free field in the Statement. AC is used through the existing analytic suppliers and implies their CC assumptions.

[F1]

A contractible space has trivial fundamental group gives the loop-homotopy hypothesis of Local weak scalar potentials and continuation on a simply connected open set, which supplies a primitive $f \in H_{l o c}^{1} (Ω)$ .

[F2]

Scalar H1 and H2 extension across a bounded Lipschitz boundary makes this bounded Lipschitz domain an $H^{1}$ extension domain, so Poincare-Wirtinger on bounded connected extension domains by Rellich compactness applies to globally $H^{1}$ functions.

[F3]

Chain rule for globally Lipschitz scalar maps of Sobolev functions applies on each relatively compact open subset and hence supplies the local chain rule. Zero weak gradient gives componentwise constants gives uniqueness.

Proof

1.1F1F2F3construct

By [F1] choose $f \in H_{l o c}^{1} (Ω)$ with $\nabla f = u$ . It remains to prove global integrability; local weak exactness alone is insufficient. Fix a ball $B ⋐ Ω$ . For $M > 0$ let $f_{M} = \max (- M, \min (f, M))$ . By [F3] locally, $f_{M} \in H_{l o c}^{1}$ , $∣ \nabla f_{M} ∣ \leq ∣ u ∣$ , and $∣ f_{M} ∣ \leq M$ . Since $Ω$ has finite measure, these local derivative identities give global $H^{1} (Ω)$ membership: the function is globally $L^{2}$ , its locally specified derivatives are globally $L^{2}$ , and every compactly supported test uses only finitely many local identities. Thus [F2] gives $∥ f_{M} - (f_{M})_{Ω} ∥_{L^{2} (Ω)} \leq C_{Ω} ∥ u ∥_{2}$ .

2.1F3step 1.1algebra∎

Put $a_{M} = (f_{M})_{Ω}$ and $b_{M} = (f_{M})_{B}$ . The preceding estimate gives $∣ a_{M} - b_{M} ∣ \leq ∣ B ∣^{- 1 / 2} C_{Ω} ∥ u ∥_{2}$ . Therefore $∥ f_{M} - b_{M} ∥_{L^{2} (Ω)} \leq C_{Ω} (1 + ∣ Ω ∣^{1 / 2} ∣ B ∣^{- 1 / 2}) ∥ u ∥_{2} .$ On $B$ , $f \in L^{2}$ and $∣ f_{M} ∣ \leq ∣ f ∣$ , so dominated convergence gives $b_{M} \to f_{B}$ . Along the integers $M \to \infty$ , $f_{M} \to f$ almost everywhere; Fatou's lemma applied to the displayed bound yields $f - f_{B} \in L^{2} (Ω)$ . Its weak gradient is $u \in L^{2}$ , hence $p = f - f_{B} \in H^{1} (Ω)$ . If two such $p$ exist, their difference has zero weak gradient, and [F3] makes it constant on connected $Ω$ . This proves every original analytic and geometric clause.

Depends on

Used by

The relative curl kernel on a contractible Lipschitz polyhedron Lemma

Dependency tree · two levels

105 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

Ralf Hiptmair and Clemens Pechstein, Regular Decompositions of Vector Fields—Continuous, Discrete, and Structure-Preserving (standard reference, not scraped)
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus: from Hodge theory to numerical stability (standard reference, not scraped)