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Subcritical compactness of the Sobolev trace

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , let $Ω \subset R^{n}$ be a bounded $C^{1}$ domain, let $T : W^{1, p} (Ω) \to L^{p} (\partial Ω)$ be the trace of The $L^{p}$ trace operator on a bounded $C^{1}$ domain, and let $1 < p < n$ with $p_{*} : = \frac{(n - 1) p}{n - p}$ . Then $T$ is compact as a map into $L^{q} (\partial Ω)$ for every $1 \leq q < p_{*}$ : every sequence bounded in $W^{1, p} (Ω)$ has a subsequence whose traces converge in $L^{q} (\partial Ω)$ . If $n < p < \infty$ , then the traces of a suitable subsequence converge in $C^{0, β} (\partial Ω)$ for every $0 \leq β < 1 - \frac{n}{p}$ , hence also in every $L^{q} (\partial Ω)$ , $1 \leq q < \infty$ ; the endpoint case $p = n$ is not claimed.

Facts & Assumptions

Given: the Axiom of Choice, a bounded $C^{1}$ domain $Ω \subset R^{n}$ , $n \geq 2$ , and $1 < p < \infty$ , $p \neq n$ , with a sequence $(u_{j})$ bounded in $W^{1, p} (Ω)$ .

[F1]

Sharp trace boundedness. For $1 < p < \infty$ and $θ = 1 - \frac{1}{p}$ , the trace satisfies $∥ T u ∥_{W^{θ, p} (\partial Ω)} \leq C ∥ u ∥_{W^{1, p} (Ω)}$ , where the boundary norm is the finite sum over a finite atlas of Euclidean $W^{θ, p}$ -norms of compactly supported chart representations, and it is independent of the atlas up to equivalence. (The sharp trace theorem: boundedness and range in the fractional space, The fractional Sobolev space on a compact $C^{1}$ boundary, Chart independence of the fractional boundary norm)

[F2]

Fractional compactness in dimension $n - 1$ . For $1 < p < n$ and $θ = 1 - \frac{1}{p}$ one has $(n - 1) - p θ = n - p > 0$ and the critical exponent of $W^{θ, p} (R^{n - 1})$ is $\frac{(n - 1) p}{n - p} = p_{*}$ ; a family of functions supported in one fixed bounded set and bounded in $W^{θ, p} (R^{n - 1})$ is therefore relatively compact in $L^{q} (R^{n - 1})$ for every $1 \leq q < p_{*}$ . (Subcritical compactness for compactly supported Slobodeckij functions, The fractional Sobolev space on a compact $C^{1}$ boundary)

[F3]

Trace and chart cutoffs. The trace commutes with multiplication by smooth ambient cutoffs and with the chart parametrisations; on a compact boundary patch the surface-measure density of the parametrisation is continuous and positive, so $L^{q}$ convergence of the finitely many chart representations gives $L^{q} (\partial Ω)$ convergence of their sum. (The trace commutes with smooth cutoffs and is chart local, Surface integration on compact C1 hypersurfaces, Finite ambient partitions near compact sets, Bounded C1 domains and their outward normals)

[F4]

The Morrey branch. For $n < p < \infty$ , the extension theorem at $k = 1$ makes the bounded $C^{1}$ domain $Ω$ a $W^{1, p}$ -extension domain; hence a bounded sequence in $W^{1, p} (Ω)$ has a subsequence whose representatives converge in $C^{0, β} (\overline{Ω})$ for every $0 \leq β < 1 - \frac{n}{p}$ , and the trace of such a class is its classical boundary restriction. (Bounded C^k domains admit integer-order Sobolev extension, Morrey--Rellich compactness for $p > n$ , The trace agrees with classical restriction for continuous Sobolev functions)

Proof

technique · bound the traces in the boundary fractional space, apply fractional compactness chart by chart, and take the Morrey branch for $p>n$

1.1F1given

Assume $1 < p < n$ . By [F1] the traces satisfy $\sup_{j} ∥ T u_{j} ∥_{W^{θ, p} (\partial Ω)} \leq C \sup_{j} ∥ u_{j} ∥_{W^{1, p} (Ω)} < \infty$ with $θ = 1 - \frac{1}{p}$ ; by the definition of the boundary norm this means that each of the finitely many compactly supported chart representations of the traces is bounded in $W^{θ, p} (R^{n - 1})$ .

2.1F2F3step 1.1given

Choose exponents $q_{ℓ} ↑ p_{*}$ with $1 \leq q_{ℓ} < p_{*}$ . For each $ℓ$ , [F2] applied successively on the finitely many charts supplies a common subsequence converging in every chart in $L^{q_{ℓ}}$ . Dependent Choice selects nested subsequences for $ℓ = 1, 2, \dots$ ; their diagonal converges in each chart for each $q_{ℓ}$ . For any $1 \leq q < p_{*}$ , choose $ℓ$ with $q < q_{ℓ}$ and use finite-measure inclusion on the common bounded chart supports. The chart Jacobian is bounded on each compact support, so [F3] transfers convergence of the finitely many chart pieces to convergence of their sum in $L^{q} (\partial Ω)$ . Thus the same subsequence works throughout the stated range.

3.1F1F2F3F4step 2.1∎

If $n < p < \infty$ , [F4] first verifies the extension-domain hypothesis and then provides a subsequence of the $u_{j}$ whose representatives converge in $C^{0, β} (\overline{Ω})$ for every $0 \leq β < 1 - \frac{n}{p}$ , and their traces, being the classical boundary restrictions, converge in $C^{0, β} (\partial Ω)$ and hence in every $L^{q} (\partial Ω)$ , $1 \leq q < \infty$ . The endpoint $p = n$ would require the limiting fractional embedding at $θ = 1 - 1 / n = d / p$ in dimension $d = n - 1$ and is deliberately not claimed. The Axiom of Choice is inherited through the published trace theorem [F1] and the Morrey branch [F4].

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Sources

Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, complete graduate notes) (standard reference, not scraped)
Eleonora Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces (arXiv:1104.4345, survey) (standard reference, not scraped)