LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Uniqueness of a weak derivative as an almost-everywhere class

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open, $n \geq 1$ , $α \in N_{0}^{n}$ , and $u, v, w \in L_{l o c}^{1} (Ω)$ . If both $v$ and $w$ are weak $α$ -derivatives of $u$ , then $v = w almost everywhere on Ω .$ Thus a weak derivative, when it exists, is unique as an almost-everywhere class. No connectedness assumption is needed.

If $Ω = \emptyset$ , the assertion holds because there is only the zero class and no nonzero tests.

Sources

Juha Kinnunen, Sobolev Spaces, Chapter 1 §1.1, printed pp. 3–4, Definition 1.2 and the uniqueness proof immediately following Remarks 1.3. The proof subtracts the two weak identities and tests their difference to conclude equality almost everywhere. The library proof uses its published regular-distribution injection as the precise uniqueness interface.
John K. Hunter, Notes on Partial Differential Equations, §3.1, Definitions 3.1–3.2, printed pp. 47–48, for the weak derivative convention.

Facts & Assumptions

Given: Countable Choice, an open set $Ω \subseteq R^{n}$ , $α \in N_{0}^{n}$ , and $u, v, w \in L_{l o c}^{1} (Ω)$ such that $v$ and $w$ both satisfy the weak derivative identity.

[F1]

A weak $α$ -derivative satisfies the signed test identity for every $φ \in C_{c}^{\infty} (Ω)$ (Weak derivative of a locally integrable function).

[F2]

Under Countable Choice, the map from a locally integrable almost-everywhere class to its regular distribution is injective (Locally integrable functions embed in distributions).

Proof

technique · direct

1.1F1given

For every test $φ \in C_{c}^{\infty} (Ω)$ , the two weak derivative identities have the same left side. Subtracting them gives $(- 1)^{∣ α ∣} \int_{Ω} (v - w) φ d x = 0.$ Since $(- 1)^{∣ α ∣}$ is nonzero, this says $\int_{Ω} (v - w) φ d x = 0$ for every test.

2.1F2step 1.1given∎

The difference $v - w$ is locally integrable, and its regular distribution therefore pairs with each test by this integral. Step 1.1 says that this regular distribution is zero. By [F2] and the stated Countable Choice hypothesis, its representing almost-everywhere class is zero; hence $v = w$ almost everywhere on all of $Ω$ . This argument applies to the whole open set, whether or not it is connected.

Depends on

Used by

Weak differentiation has a closed graph on its natural domains Corollary
Cantor function has singular distributional derivative Counterexample
Subcritical W^1,p is not closed under multiplication Counterexample
Integer-order Sobolev spaces and their norms Definition
Matching C¹ pieces across a hyperplane have no jump derivative Example
Sharp Sobolev threshold for a radial power Example
Classical derivatives agree with weak derivatives Lemma
Integration by parts for dual-exponent Sobolev functions Lemma
Linearity, locality, and commutation of weak derivatives Lemma
Sobolev functions paste across an overlap Lemma
The Sobolev norm descends to equivalence classes Lemma
Weak derivatives persist under local Lp limits Lemma
Weak Leibniz rule with a smooth factor Lemma
Weak derivatives are represented distributional derivatives Remark
Integer-order W^k,2 and Hᵏ agree with equivalent norms Theorem
The ACL characterisation of W^1,p Theorem

Dependency tree · two levels

19 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

Juha Kinnunen, Sobolev Spaces (2026), Chapter 1 §1.1 (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014), §3.1 (standard reference, not scraped)