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.

Uniqueness of a weak derivative as an almost-everywhere class

Statement

Sources

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources