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Rellich--Kondrachov at the critical source exponent $p = n$

Statement

Assume the Axiom of Choice. Let $n \geq 2$ and let $Ω \subseteq R^{n}$ be a bounded extension domain. Then $W^{1, n} (Ω)$ is compactly embedded in $L^{q} (Ω)$ for every finite $q$ : for each fixed $1 \leq q < \infty$ , every sequence bounded in $W^{1, n} (Ω)$ has a subsequence converging in $L^{q} (Ω)$ . There is no claim of compactness into $L^{\infty}$ .

Facts & Assumptions

Given: the Axiom of Choice, a bounded extension domain $Ω \subseteq R^{n}$ , $n \geq 2$ , and a sequence $(u_{j})$ with $M : = \sup_{j} ∥ u_{j} ∥_{W^{1, n} (Ω)} < \infty$ .

[F1]

$L^{n}$ -compactness. Some subsequence converges in $L^{n} (Ω)$ . (Compactness of $W^{1, p} (Ω) ↪ L^{p} (Ω)$ on bounded extension domains, Sobolev extension domains and extension operators, Compactly embedded normed spaces)

[F2]

Critical embedding into every finite $L^{q^{'}}$ . For every finite $q^{'} > 1$ there is $C (q^{'})$ with $∥ v ∥_{L^{q^{'}} (Ω)} \leq C (q^{'}) ∥ v ∥_{W^{1, n} (Ω)}$ for all $v \in W^{1, n} (Ω)$ ; the sequence is therefore uniformly bounded in $L^{q^{'}} (Ω)$ for each fixed finite $q^{'}$ . (Higher-order Sobolev embedding (case $k = 1$ , $p = n$ ), Integer-order Sobolev spaces and their norms)

[F3]

Interpolation and H"older. For $n \leq b < q^{'}$ , $∥ g ∥_{b} \leq ∥ g ∥_{n}^{θ} ∥ g ∥_{q^{'}}^{1 - θ}$ with $1 / b = θ / n + (1 - θ) / q^{'}$ ; for $b \leq n$ , $∥ g ∥_{L^{b} (Ω)} \leq ∣ Ω ∣^{1 / b - 1 / n} ∥ g ∥_{L^{n} (Ω)}$ . (Lyapunov interpolation inequality for $L^{p}$ norms, Holder's inequality for integrals, including the endpoint cases, The space $L^{p} (μ)$ as the quotient by null functions)

[F4]

Under Countable Choice, each $L^{q} (Ω)$ , $1 \leq q < \infty$ , is complete. (Riesz-Fischer completeness of $L^{p}$ for $1 \leq p \leq \infty$ )

Proof

technique · direct

1.1F1F2given

If $Ω = \emptyset$ , all classes are zero and the claim is immediate. Otherwise fix $1 \leq q < \infty$ and choose $q^{'} > \max {q, n}$ . By [F1] extract a subsequence converging in $L^{n} (Ω)$ and write $w_{k} : = u_{j_{k}}$ ; by [F2] the differences satisfy $∥ w_{k} - w_{ℓ} ∥_{L^{q^{'}} (Ω)} \leq 2 C (q^{'}) M$ .

2.1F1F3F4step 1.1

If $q \leq n$ , then $∥ w_{k} - w_{ℓ} ∥_{L^{q}} \leq ∣ Ω ∣^{1 / q - 1 / n} ∥ w_{k} - w_{ℓ} ∥_{L^{n}} \to 0$ by [F3] and step 1.1. If $q > n$ , then $n < q < q^{'}$ and [F3] gives $∥ w_{k} - w_{ℓ} ∥_{L^{q}} \leq ∥ w_{k} - w_{ℓ} ∥_{L^{n}}^{θ} (2 C (q^{'}) M)^{1 - θ} \to 0$ for the corresponding $θ \in (0, 1)$ . In both cases $(w_{k})$ is Cauchy, hence convergent by [F4], in $L^{q} (Ω)$ .

3.1F1F2step 1.1step 2.1∎

Every bounded sequence in $W^{1, n} (Ω)$ therefore has a subsequence converging in $L^{q} (Ω)$ for each fixed finite $q$ , so $W^{1, n} (Ω) ⋐ L^{q} (Ω)$ in the sense of Compactly embedded normed spaces. No compactness into $L^{\infty}$ is asserted. The proof uses only finite target exponents. The Axiom of Choice is inherited through [F1] and the supplier [F2].

Depends on

Used by

Dependency tree · two levels

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (archived 2025 author manuscript) (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)
Juha Kinnunen, Sobolev Spaces (Aalto University, 2026, complete graduate lecture notes) (standard reference, not scraped)

Rellich--Kondrachov at the critical source exponent p=np=n