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Zero mean of the divergence under zero normal trace

Statement

Assume Countable Choice. Let $D \subset R^{d}$ be bounded and open. If $v \in H_{0} (div; D)$ , then $div v \in L^{1} (D)$ and $\int_{D} div v d x = 0.$

Facts & Assumptions

Given: Countable Choice, a bounded open set $D$ , and $v$ with zero weak normal trace.

[A1]

Countable Choice is inherited from the Sobolev and trace-quotient interfaces. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

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

[F2]

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

[F3]

Bounded sets have finite Lebesgue measure, so the constant function $1$ belongs to $L^{2} (D)$ and has zero weak gradient in $H^{1} (D)$ . (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure, Integer-order Sobolev spaces and their norms)

[F4]

Holder's inequality implies $∥ g ∥_{L^{1} (D)} \leq ∣ D ∣^{1 / 2} ∥ g ∥_{L^{2} (D)}$ for $g \in L^{2} (D)$ . (Holder's inequality for integrals, including the endpoint cases)

[F5]

The divergence of an H(div) field belongs to $L^{2} (D)$ . (The real H(div) space)

Proof

1.1A1F3F4F5construct

Since $div v \in L^{2} (D)$ by [F5], and $∣ D ∣ < \infty$ by [F3], [F4] shows $div v \in L^{1} (D)$ . The constant class $1$ is an admissible element of $H^{1} (D)$ by [F3] and the definition of the weak Sobolev norm, so its trace class $[1]$ lies in $T_{D}$ .

2.1F2step 1.1algebra

Apply [F2] with $w = 1$ . Since $\nabla 1 = 0$ , Green duality gives $⟨ γ_{n} v, [1] ⟩ = \int_{D} div v d x$ .

3.1F1step 2.1∎

By [F1], $γ_{n} v = 0$ as an element of $T_{D}^{'}$ acting on $T_{D}$ , so its value at $[1] \in T_{D}$ is zero. The identity in step 2.1 therefore gives $\int_{D} div v = 0$ .

Depends on

Used by

Dependency tree · two levels

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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)