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Compactness of $W^{1, p} (Ω) ↪ L^{p} (Ω)$ on bounded extension domains

Statement

Assume the Axiom of Choice. Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be a bounded $W^{1, p}$ -extension domain (Sobolev extension domains and extension operators) and let $1 \leq p < \infty$ . Then $W^{1, p} (Ω)$ is compactly embedded in $L^{p} (Ω)$ : every sequence bounded in $W^{1, p} (Ω)$ has a subsequence converging in $L^{p} (Ω)$ . Every bounded $C^{1}$ domain is an example.

Facts & Assumptions

Given: the Axiom of Choice, a bounded $W^{1, p}$ -extension domain $Ω \subseteq R^{n}$ , $1 \leq p < \infty$ , and a sequence $(u_{j})$ with $M : = \sup_{j} ∥ u_{j} ∥_{W^{1, p} (Ω)} < \infty$ .

[F1]

Extension operator. There is a bounded linear $E : W^{1, p} (Ω) \to W^{1, p} (R^{n})$ with $(E u) ∣_{Ω} = u$ and $∥ E u ∥_{W^{1, p} (R^{n})} \leq ∥ E ∥ ∥ u ∥_{W^{1, p} (Ω)}$ . (Sobolev extension domains and extension operators)

[F2]

Cutoff. Since $\overline{Ω}$ is compact and contained in the open set $V : = B (0, R)$ , with $R > 0$ large enough to contain $\overline{Ω}$ , there is $η \in C_{c}^{\infty} (R^{n})$ with $η = 1$ on $\overline{Ω}$ and $supp η \subseteq V$ , a fixed bounded set. This construction also works for the empty domain, choosing any ball $V$ . (A Euclidean bump for a compact set inside an open set, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space)

[F3]

Products with a smooth cutoff. If $v \in W^{1, p} (R^{n})$ and $η \in C_{c}^{\infty} (R^{n})$ , then $η v \in W^{1, p} (R^{n})$ with $∥ η v ∥_{W^{1, p} (R^{n})} \leq C_{η} ∥ v ∥_{W^{1, p} (R^{n})}$ for a constant depending only on $η$ and $p$ . (Weak Leibniz rule with a smooth factor)

[F4]

Translation estimate and automatic tails. Under the Axiom of Choice, $∥ τ_{h} v - v ∥_{L^{p} (R^{n})} \leq ∣ h ∣ ∥ ∣ D v ∣ ∥_{L^{p} (R^{n})}$ ; a family supported in one fixed bounded set has vanishing tails. (The translation estimate for $W^{1, p}$ functions on $R^{n}$ , Uniformly supported families have vanishing tails)

[F5]

The Fr'echet--Kolmogorov criterion. A bounded family in $L^{p} (R^{n})$ with vanishing tails and uniform translation control has an $L^{p} (R^{n})$ -convergent subsequence. (The Fr'echet--Kolmogorov compactness criterion in $L^{p} (R^{n})$ , The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain)

[F6]

Restriction and norms. $∥ g ∣_{Ω} ∥_{L^{p} (Ω)} \leq ∥ g ∥_{L^{p} (R^{n})}$ ; and $(E u_{j}) ∣_{Ω} = u_{j}$ almost everywhere. (Integer-order Sobolev spaces and their norms, The space $L^{p} (μ)$ as the quotient by null functions)

[F7]

Bounded $C^{1}$ domains are extension domains. (Bounded C^k domains admit integer-order Sobolev extension)

Proof

technique · Extend, multiply by a fixed cutoff, apply the Fr\'echet--Kolmogorov criterion to the products, and restrict to $\Omega$

1.1F1F2F3F6given

Fix $E$ as in [F1] and $η$ as in [F2], and put $v_{j} : = η E u_{j}$ . By [F3] each $v_{j}$ lies in $W^{1, p} (R^{n})$ and is supported in the fixed bounded set $supp η$ ; moreover $∥ v_{j} ∥_{W^{1, p} (R^{n})} \leq C_{η} ∥ E u_{j} ∥_{W^{1, p} (R^{n})} \leq C_{η} ∥ E ∥ M$ , and $v_{j} = (E u_{j}) ∣_{Ω} = u_{j}$ almost everywhere on $Ω$ because $η = 1$ there.

2.1F4F5step 1.1algebra

The family ${v_{j}}$ satisfies the three hypotheses of [F5]: it is bounded in $L^{p} (R^{n})$ by step 1.1; its tails vanish by [F4] because all members are supported in the fixed bounded set $supp η$ ; and [F4] gives $∥ τ_{h} v_{j} - v_{j} ∥_{L^{p} (R^{n})} \leq ∣ h ∣ ∥ ∣ D v_{j} ∣ ∥_{L^{p} (R^{n})} \leq C^{'} M^{'} ∣ h ∣$ with $C^{'}$ and $M^{'}$ independent of $j$ , so the translation control is uniform and tends to $0$ with $∣ h ∣$ .

3.1F1F5F6F7step 1.1step 2.1∎

By [F5] some subsequence $(v_{j_{k}})$ converges in $L^{p} (R^{n})$ , say to $v$ ; restricting and using $v_{j_{k}} = u_{j_{k}}$ almost everywhere on $Ω$ together with [F6] gives $∥ u_{j_{k}} - v ∣_{Ω} ∥_{L^{p} (Ω)} = ∥ (v_{j_{k}} - v) ∣_{Ω} ∥_{L^{p} (Ω)} \leq ∥ v_{j_{k}} - v ∥_{L^{p} (R^{n})} \to 0$ , so $(u_{j_{k}})$ converges in $L^{p} (Ω)$ . This proves compactness of the inclusion; its boundedness follows from $∥ u ∥_{L^{p} (Ω)} \leq ∥ E u ∥_{L^{p} (R^{n})} \leq ∥ E ∥ ∥ u ∥_{W^{1, p} (Ω)}$ by [F1] and [F6]. Finally, [F7] says every bounded $C^{1}$ domain carries such an extension operator, giving the stated examples. The Axiom of Choice is inherited through [F1], [F4] and [F7], while the extraction uses the Countable and Dependent Choice of [F5].

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Sources

Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, complete graduate notes) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter notes) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (archived 2025 author manuscript) (standard reference, not scraped)

Compactness of W1,p(Ω)↪Lp(Ω)W1,p(Ω)↪Lp(Ω) on bounded extension domains