ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02

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Compactly supported Sobolev functions extend by zero without a jump

Example

Assume Countable Choice. Let $n \geq 1$ and $Ω \subseteq R^{n}$ be open, $1 \leq p \leq \infty$ and $K \in {R, C}$ . If $u \in W^{1, p} (Ω; K)$ vanishes almost everywhere outside a compact set $K_{0} \subset Ω$ , then its extension by zero belongs to $W^{1, p} (R^{n}; K)$ , its first weak derivatives are the zero extensions of the weak derivatives $D_{i} u$ , and $∥ E_{0} u ∥_{W^{1, p} (R^{n})} = ∥ u ∥_{W^{1, p} (Ω)} .$ Compact support inside $Ω$ is what makes this work: the extension has no jump at $\partial Ω$ , in contrast with the indicator of $(0, 1)$ of the companion page. Nothing here asserts membership of $u$ in $W_{0}^{1, \infty} (Ω)$ , which is a statement about approximation by test functions, not about extension.

Facts & Assumptions

Given: Countable Choice; an open set $Ω \subseteq R^{n}$ ; $1 \leq p \leq \infty$ ; $K \in {R, C}$ ; and a class $u \in W^{1, p} (Ω; K)$ with a representative vanishing almost everywhere outside a compact set $K_{0} \subset Ω$ .

[L1]

Compactly supported Sobolev classes extend by zero in every integer order $k$ : under the stated hypotheses on $Ω$ , $p$ , $K$ and $k$ , if $u \in W^{k, p} (Ω; K)$ vanishes almost everywhere outside a compact $K_{0} \subset Ω$ , then $E_{0} u \in W^{k, p} (R^{n}; K)$ , $D^{α} (E_{0} u) = E_{0} (D^{α} u)$ almost everywhere for $∣ α ∣ \leq k$ , and $∥ E_{0} u ∥_{W^{k, p} (R^{n})} = ∥ u ∥_{W^{k, p} (Ω)}$ (Compactly supported Sobolev functions extend by zero in every integer order).

[L2]

For $1 \leq p < \infty$ , $∥ w ∥_{W^{1, p} (Ω)} = (∥ w ∥_{L^{p} (Ω)}^{p} + \sum_{i < n} ∥ D_{i} w ∥_{L^{p} (Ω)}^{p})^{1 / p}$ ; at $p = \infty$ the norm is $\max_{∣ α ∣ \leq 1} ∥ D^{α} w ∥_{L^{\infty} (Ω)}$ , and likewise on $R^{n}$ (Integer-order Sobolev spaces and their norms).

[L3]

$W_{0}^{1, \infty} (Ω)$ is the closure of $C_{c}^{\infty} (Ω)$ in the $W^{1, \infty}$ norm; membership is a density statement about test functions, not about extension or support (Zero-boundary Sobolev space as a norm closure).

Verification

technique · direct

1.1L1given

The hypotheses of [L1] with $k = 1$ hold: $u \in W^{1, p} (Ω; K)$ vanishes almost everywhere outside the compact set $K_{0} \subset Ω$ , and $1 \leq p \leq \infty$ with the same scalar field.

2.1L1L2step 1.1

Applying [L1] with $k = 1$ : the zero extension $E_{0} u$ lies in $W^{1, p} (R^{n}; K)$ , its first weak derivatives are $D_{i} (E_{0} u) = E_{0} (D_{i} u)$ almost everywhere, and the norms agree, so $∥ E_{0} u ∥_{W^{1, p} (R^{n})} = ∥ u ∥_{W^{1, p} (Ω)}$ by [L2].

3.1L3step 2.1given∎

Scope. The conclusion is an extension statement for the class of $u$ ; it uses only compact essential support and $W^{1, p}$ regularity and yields no membership in $W_{0}^{1, \infty} (Ω)$ , which by [L3] would require approximating $u$ by test functions in the $W^{1, \infty}$ norm.

Depends on

Used by

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Dependency tree · two levels

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Sources

Juha Kinnunen, Sobolev Spaces (2026), Lemma 1.14(4)–(5) and Theorem 1.25 (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014), §3.4 (standard reference, not scraped)