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A nonzero boundary value creates a zero-extension jump

Statement refuted

The claim that unrestricted zero extension is a Sobolev extension operator — that for every open $Ω \subseteq R^{n}$ and every $1 \leq p \leq \infty$ the extension of a class $u \in W^{1, p} (Ω; K)$ by zero outside $Ω$ belongs to $W^{1, p} (R^{n}; K)$ — is false. On $Ω = (0, 1)$ the constant $u \equiv 1$ belongs to $W^{1, p} (Ω)$ for every $1 \leq p \leq \infty$ , while its zero extension $χ_{(0, 1)}$ has distributional derivative $δ_{0} - δ_{1}$ and does not belong to $W^{1, p} (R)$ for any $p$ . Consequently zero extension is not an extension operator in the sense of Sobolev extension domains and extension operators.

Facts & Assumptions

Given: Countable Choice; $Ω = (0, 1)$ ; the constant function $u \equiv 1$ on $Ω$ ; its zero extension $H = χ_{(0, 1)}$ on $R$ ; the Dirac distributions $δ_{0}, δ_{1}$ ; and $1 \leq p \leq \infty$ .

[F1]

A locally integrable $v$ is the weak first derivative of a locally integrable $u$ on an open set exactly when $\int u φ^{'} = - \int v φ$ for every test $φ$ ; this is equivalent to the distributional identity $\partial T_{u} = T_{v}$ , and weak derivatives are unique up to almost-everywhere equality (Weak derivative of a locally integrable function).

[F2]

A weak $L_{l o c}^{1}$ derivative of a locally integrable $u$ exists exactly when the distributional derivative $\partial T_{u}$ is a regular distribution $T_{v}$ with $v \in L_{l o c}^{1}$ ; in that case $v$ is unique almost everywhere, and distributional derivatives of general distributions need not be regular (Weak derivatives are represented distributional derivatives).

[F3]

The Dirac distribution at $a$ is defined by $⟨ δ_{a}, ψ ⟩ = ψ (a)$ ; for a $C^{1}$ function on $[0, 1]$ $⟨ T_{χ_{(0, 1)}}, ψ^{'} ⟩ = \int_{0}^{1} ψ^{'}$ and $⟨ \partial T_{χ_{(0, 1)}}, ψ ⟩ = - \int_{0}^{1} ψ^{'}$ , and $\int_{0}^{1} ψ^{'} = ψ (1) - ψ (0)$ (Dirac delta and its derivatives, Complex integration by parts on intervals and decaying lines).

[F4]

Intervals in $R$ are Lebesgue measurable with their length as measure, and singletons are null; in particular $χ_{(0, 1)} \in L^{p} (R)$ for every $1 \leq p \leq \infty$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, Complex Lp classes and Euclidean test-function conventions).

[F5]

For $0 < r < R$ and $n \geq 1$ there is a smooth cut-off equal to one on the closed ball of radius $r$ with support in the open ball of radius $R$ ; after translation and scaling this gives, for each integer $m \geq 1$ , a $[0, 1]$ -valued $ψ_{m} \in C_{c}^{\infty} (R)$ with $ψ_{m} (0) = 1$ and $supp ψ_{m} \subseteq (- 1 / m, 1 / m)$ (A smooth bump between concentric Euclidean balls).

[F6]

Dominated convergence: if $g_{m} \to g$ almost everywhere and $∣ g_{m} ∣ \leq G$ for a single integrable $G$ , then $\int g_{m} \to \int g$ (Dominated convergence).

[F7]

Each $W^{1, p} (Ω; K)$ class lies in $L^{p} (Ω; K)$ and has weak coordinate derivatives in $L^{p} (Ω; K)$ , and $L^{p} \subseteq L_{l o c}^{1}$ on a $σ$ -finite open set (Integer-order Sobolev spaces and their norms).

Choice use. The declared interfaces of [F1], [F2] and [F7] are those of the published Sobolev and distributional calculi, which assume Countable Choice; the concrete computation of the distributional derivative and the approximating cut-offs is explicit.

Counterexample

1.1F1F3F7given

The constant $u \equiv 1$ satisfies $u \in L^{p} ((0, 1))$ for every $1 \leq p \leq \infty$ , since $\int_{0}^{1} ∣ 1 ∣^{p} = 1 < \infty$ and $∣ 1 ∣$ is bounded. For every $φ \in C_{c}^{\infty} ((0, 1))$ the functions $φ$ and the constant $0$ give $\int_{0}^{1} u φ^{'} = \int_{0}^{1} φ^{'} = φ (1) - φ (0) = 0 = - \int_{0}^{1} 0 \cdot φ,$ because $φ$ is supported in $(0, 1)$ . By [F1] the zero class is the weak derivative of $u$ , so $u \in W^{1, p} ((0, 1))$ for every $p$ by [F7].

1.2F1F3F4given

The zero extension $H = χ_{(0, 1)}$ is measurable with $∣ H ∣ \leq 1$ , so $H \in L^{p} (R)$ for every $p$ by [F4]. For $φ \in C_{c}^{\infty} (R)$ the distributional derivative acts by $⟨ \partial T_{H}, φ ⟩ = - \int_{R} H φ^{'} = - \int_{0}^{1} φ^{'} = - (φ (1) - φ (0)) = ⟨ δ_{0} - δ_{1}, φ ⟩,$ so $\partial T_{H} = δ_{0} - δ_{1}$ in $D^{'} (R)$ .

1.3F3F5given

Fix the cut-offs $ψ_{m}$ of [F5]. They satisfy $ψ_{m} (0) = 1$ , $ψ_{m} (1) = 0$ for $m \geq 1$ , and $⟨ δ_{0} - δ_{1}, ψ_{m} ⟩ = ψ_{m} (0) - ψ_{m} (1) = 1 - 0 = 1$ for every $m \geq 1$ .

2.1F2step 1.3

Suppose, for contradiction, that $\partial T_{H}$ were a regular distribution $T_{v}$ with $v \in L_{l o c}^{1} (R)$ , that is, that $H$ had a weak derivative in $L_{l o c}^{1} (R)$ . Testing the identity $T_{v} = δ_{0} - δ_{1}$ against each $ψ_{m}$ of step 1.3 gives $\int_{R} v ψ_{m} = ⟨ δ_{0} - δ_{1}, ψ_{m} ⟩ = 1$ for every $m \geq 1$ .

3.1F2F6step 2.1

On the other hand $v ψ_{m} \to 0$ almost everywhere and $∣ v ψ_{m} ∣ \leq ∣ v ∣ χ_{(- 1, 1)}$ , which is integrable because $v$ is locally integrable; [F6] therefore gives $\int_{R} v ψ_{m} \to 0$ . This contradicts step 2.1. Hence $\partial T_{H}$ is not a regular distribution, and by [F2] the function $H$ has no weak first derivative in $L_{l o c}^{1} (R)$ .

4.1F7step 1.1step 3.1given∎

If $H$ belonged to $W^{1, p} (R)$ for some $1 \leq p \leq \infty$ , then by [F7] its weak first derivative would be an $L^{p}$ class, hence in particular a weak $L_{l o c}^{1}$ derivative, contradicting step 3.1. Therefore $H \notin W^{1, p} (R)$ for every $p$ , while $H ∣_{Ω} = u$ and $u \in W^{1, p} (Ω)$ by step 1.1. The map $u \mapsto χ_{(0, 1)} u$ therefore fails to send $W^{1, p} (Ω)$ into $W^{1, p} (R)$ for each exponent, so it is not an extension operator in the sense of Sobolev extension domains and extension operators.

Depends on

Used by

Mollifying a zero extension leaks across the boundary Counterexample

Dependency tree · two levels

74 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), Theorem 1.25 and Example 1.7 (standard reference, not scraped)
Richard S. Laugesen, Linear Analysis and Partial Differential Equations (2020), §3.6 (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014), Example 3.4 (standard reference, not scraped)