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Classical derivatives agree with weak derivatives

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open, $n \geq 1$ , let $k \in N_{0}$ , and let $u : Ω \to C$ have real and imaginary parts of class $C^{k}$ . For each multi-index $α \in N_{0}^{n}$ with $∣ α ∣ \leq k$ , the componentwise classical derivative $\partial^{α} u$ is locally integrable and is the weak derivative $D^{α} u$ . By the uniqueness lemma, it represents the unique locally integrable weak-derivative class. For $α = 0$ , this says $D^{0} u = u$ .

If $Ω = \emptyset$ , the assertion is vacuous: there is only the zero class and every displayed test identity is zero.

Facts & Assumptions

Given: Countable Choice, an open set $Ω \subseteq R^{n}$ , $u \in C^{k} (Ω; C)$ , $k \in N_{0}$ , and a multi-index $α$ with $∣ α ∣ \leq k$ .

[F1]

By the regular-distribution pairing and signed-transpose convention, the weak test identity is equivalent to $\partial^{α} T_{f} = T_{g}$ as distributions (Weak derivative of a locally integrable function).

[F2]

Under Countable Choice, classical derivatives of $C^{k}$ functions are compatible with distributional derivatives: Distributional differentiation is continuous and commutes.

[F3]

Under Countable Choice, a locally integrable weak derivative is unique almost everywhere (Uniqueness of a weak derivative as an almost-everywhere class).

Proof

technique · direct

1.1F2given

Apply [F2] to $u$ and $α$ . The regular distributions of $u$ and $\partial^{α} u$ are defined, so in particular both functions are locally integrable, and $\partial^{α} T_{u} = T_{\partial^{α} u} .$ The Countable Choice use is exactly the comparison of the compactly supported Riemann integrals with the corresponding Lebesgue integrals in [F2].

2.1F1step 1.1given

By [F1], this distributional equality is equivalent to the weak test identity. Explicitly, evaluation on any $φ \in C_{c}^{\infty} (Ω)$ gives $(- 1)^{∣ α ∣} \int_{Ω} u D^{α} φ d x = \int_{Ω} (\partial^{α} u) φ d x .$ Multiplying by $(- 1)^{∣ α ∣}$ gives the defining weak identity for $\partial^{α} u$ ; the sign is its own inverse. Since the test was arbitrary, $\partial^{α} u$ is a weak derivative.

3.1F3step 2.1given∎

The weak derivative class is unique by [F3], so the locally integrable class represented by $\partial^{α} u$ is the class denoted $D^{α} u$ . When $α = 0$ , the identity reduces to $u = u$ .

Sources

Juha Kinnunen, Sobolev Spaces, Chapter 1 §1.1, printed pp. 2–3: compact-support integration by parts has no boundary term, successive integrations give the multi-index identity, and Remarks 1.3(1) state that classical derivatives through order $k$ are weak derivatives.
John K. Hunter, Notes on Partial Differential Equations, Chapter 3 §3.1, printed pp. 47–48, for the weak-derivative integration-by-parts convention.

Classical derivatives agree with weak derivatives

Statement

Facts & Assumptions

Proof

Sources

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Sources