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H1-zero fields have zero weak H(div) normal trace

Statement

Assume Countable Choice and let $D \subset R^{d}$ be open. Every real vector field $w \in H_{0}^{1} (D; R^{d})$ belongs to $H_{0} (div; D)$ , where the latter is the kernel of the weak normal trace. In particular, $div w \in L^{2} (D)$ and its weak normal trace vanishes on the whole boundary trace quotient.

Facts & Assumptions

Given: Countable Choice, an open set $D$ , and $w \in H_{0}^{1} (D; R^{d})$ .

[A1]

Countable Choice selects the compactly supported smooth approximants in the H1-zero closure. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

Each component of $H^{1} (D; R^{d})$ has its first weak derivatives in $L^{2} (D)$ ; their finite sum is the distributional divergence. (Integer-order Sobolev spaces and their norms, The real H(div) space)

[F2]

$H_{0}^{1} (D)$ is the $H^{1}$ -norm closure of $C_{c}^{\infty} (D)$ . (Zero-boundary Sobolev space as a norm closure)

[F3]

Green duality defines the weak normal trace by $⟨ γ_{n} w, [ϕ] ⟩ = \int_{D} (w \cdot \nabla ϕ + (div w) ϕ)$ . (Weak normal trace by Green duality)

[F4]

$H_{0} (div; D)$ is exactly the kernel of this weak normal trace. (Zero-normal-trace H(div) space)

[F5]

Cauchy–Schwarz controls each volume pairing in Green duality. (Cauchy-Schwarz inequality for $L^{2}$ )

Proof

1.1F1constructalgebra

Since each component of $w$ and each first weak derivative is in $L^{2} (D)$ by [F1], the finite sum $div w = \sum_{j = 1}^{d} \partial_{j} w_{j}$ lies in $L^{2} (D)$ . Thus $w \in H (div; D)$ by its definition.

2.1A1F2F3F4F5choosealgebra∎

By [A1,F2], choose vector fields $w_{n} \in C_{c}^{\infty} (D; R^{d})$ with $w_{n} \to w$ in $H^{1}$ . For arbitrary $ϕ \in H^{1} (D)$ , the weak-derivative identity for $ϕ$ , tested against each compactly supported component $(w_{n})_{j}$ , gives $\int_{D} w_{n} \cdot \nabla ϕ + \int_{D} (div w_{n}) ϕ = 0.$ The H1 convergence implies $w_{n} \to w$ in $L^{2}$ and $div w_{n} \to div w$ in $L^{2}$ . Specifically, [F5] bounds the difference of the Green pairings by $∥ w_{n} - w ∥_{2} ∥ \nabla ϕ ∥_{2} + ∥ div w_{n} - div w ∥_{2} ∥ ϕ ∥_{2} ⟶ 0.$ Thus $\int_{D} w \cdot \nabla ϕ + \int_{D} (div w) ϕ = 0$ . This holds for every $ϕ \in H^{1} (D)$ ; hence [F3] gives $γ_{n} w = 0$ , and [F4] implies $w \in H_{0} (div; D)$ .

Depends on

Used by

Continuous mixed Darcy well-posedness with pressure gauge Theorem

Dependency tree · two levels

25 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

Daniele Boffi, Franco Brezzi and Michel Fortin, Mixed Finite Element Methods and Applications (standard reference, not scraped)
Jean-Luc Guermond, MATH 661 Fall 2025, Part III Chapter 14: H(div) finite elements (standard reference, not scraped)