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Zero mean of the divergence of a zero-boundary field

Statement

Assume Countable Choice. Let $D \subset R^{d}$ be bounded and open. If $v \in H_{0}^{1} (D; R^{d})$ , then $div v \in L^{2} (D)$ and $\int_{D} div v = 0$ .

Facts & Assumptions

Given: A bounded open $D$ and a vector field in the $H^{1}$ -closure of $C_{c}^{\infty} (D; R^{d})$ .

[A1]

Countable Choice permits selecting the H0 test-function approximation sequence used to pass the integral identity to the limit. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

$H_{0}^{1} (D)$ is the closure of compactly supported smooth functions in the $W^{1, 2}$ norm. (Zero-boundary Sobolev space as a norm closure)

[F2]

First weak derivatives of a $W^{1, 2}$ field lie in $L^{2}$ , and its $W^{1, 2}$ norm controls their $L^{2}$ norms. (Integer-order Sobolev spaces and their norms)

[F3]

On a finite-measure set, $∣ \int_{D} g ∣ \leq ∣ D ∣^{1 / 2} ∥ g ∥_{2}$ . (Holder's inequality for integrals, including the endpoint cases)

[F4]

Fubini evaluates the integral of each compactly supported coordinate derivative by iterated one-dimensional integration. (Fubini's theorem for L^1 functions on a sigma-finite product)

Proof

1.1F2algebra

By [F2], each weak derivative $\partial_{i} v_{i}$ is in $L^{2}$ , so their finite sum $div v$ is in $L^{2} (D)$ .

1.2givenF1F4

For $v \in C_{c}^{\infty} (D; R^{d})$ , each component derivative has integral zero by one-dimensional integration along its coordinate, hence $\int_{D} div v = 0$ .

2.1A1F1F2F3step 1.2∎

By [A1, F1], choose $v_{n} \in C_{c}^{\infty} (D; R^{d})$ converging to $v$ in $H^{1}$ . Then $div v_{n} \to div v$ in $L^{2}$ by [F2]. Applying [F3] to the difference shows their integrals converge; the zero integrals in step 1.2 therefore give $\int_{D} div v = 0$ .

Depends on

Used by

Dependency tree · two levels

34 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

Jean-Luc Guermond, MATH 661 Fall 2025, Chapters 53–54 (complete course notes) (standard reference, not scraped)
Daniele Boffi, Franco Brezzi, Michel Fortin, Mixed Finite Element Methods and Applications (complete textbook) (standard reference, not scraped)