Trace Theorems

The spaces just introduced give us the key to take traces of functions 𝐹 ∶ U(Ϻ) → ℝ on ESMs. These results are usually given for Ω ∈ 𝕃𝕚𝕡_𝑑 or ℝ𝑑 and Ω𝑘= Ω ∩ (ℝ𝑘_× {0}𝑑−𝑘) or ℝ𝑘× {0}𝑑−𝑘, but since we have a finite set of Lipschitz parameter spaces and C-bounded parameterisations, they generalise directly to ESMs.

To this end, we define now the restriction Ͳ_Ϻ ∶ C∞(ℝ𝑑_{) → C}∞_{(Ϻ) pointwise and}

for the fractional and integer Sobolev spaces via completion. Then we obtain the following trace theorems, for which we present literature references and proofs of certain generalising aspects in Sect. 9.2.5 of the appendix.

2.31 Theorem — Integer Trace Theorem —

1. Let Ϻ ∈ 𝕄𝑘_bd(ℝ𝑑_{) be equipped with finite inverse atlas 𝔸}

Ϻ = {(ψ𝑖, ω𝑖)}𝑖∈𝐼 ∈ ℿ(Ϻ) and let U(Ϻ) ∈ 𝕃𝕚𝕡_𝑑 be some ambient tubular neighbourhood of Ϻ with

fixed extent 𝜚 > 0. Let 𝐹 ∈ W𝑚_𝑝(U(Ϻ)) for some 1 ≤ 𝑝 <

∞

Ͳ_Ϻ𝐹 ∈ W𝜇𝑝(Ϻ) for any 𝜇 ∈ ℕ0 with 𝜇 < 𝑚 −𝑑−𝑘𝑝 (𝜇 ≤ 𝑚 −𝑑−𝑘𝑝 in case 𝑝 = 1) and ‖Ͳ_Ϻ𝐹‖_W𝜇

𝑝(Ϻ)≤ c ‖𝐹‖W 𝑚 𝑝(U(Ϻ)).

2. Let 𝑓 ∈ W𝑚_𝑝(Ϻ) for some Ϻ ∈ 𝕄𝑘

bd(ℝ𝑑) and some 1 ≤ 𝑝 <

∞

some open and bounded Ϻ₀ ∈ 𝕄𝑘_bd(ℝ𝑑) with Ϻ0 ⋐ Ϻ that has nonempty smooth

boundary Γ. Then ͲΓ𝑓 ∈ W

𝜇

𝑝(Γ) for any 𝜇 < 𝑚 and ‖ͲΓ𝑓‖W𝜇_𝑝(Γ)≤ c ‖𝑓‖W𝑚_𝑝(Ϻ0).

Things become more involved when one is willing to consider fractional orders. The gain we can hope for in that setting is that if we consider fractional spaces, the loss of regularity might be considerably lower than implied by the integer case, and so we could hope for tighter relations in future approximation results.

Unfortunately, taking traces of Sobolev spaces will not necessarily bring you into a Sobolev space as the most regular space to end up in, not even a fractional one. But there is one important exception to this: Whenever 𝑝 = 2, then everything coincides and we are in a Slobodeckij space. So this is the case we go for:

2.32 Theorem — Fractional Trace Theorem —

Let Ϻ ∈ 𝕄𝑘_bd(ℝ𝑑) have finite inverse atlas 𝔸Ϻ = {(ψ𝑖, ω𝑖)}𝑖∈𝐼 ∈ ℿ(Ϻ) and let

U(Ϻ) ∈ 𝕃𝕚𝕡_𝑑 be some tubular neighbourhood of Ϻ with fixed extent 𝜚 > 0 and ex-

tended inverse atlas {(Ψ_𝑖, Ω_𝑖)}_𝑖∈𝐼∈ ℿex

N(Ϻ) subordinate to 𝔸Ϻ. Let 𝐹 ∈ H 𝑟 (U(Ϻ)) for 𝑟 > 𝑑−𝑘 2 . Then ͲϺ𝐹 ∈ H 𝜚 (Ϻ) for 𝜚 = 𝑟 −𝑑−𝑘 2 > 0 and ‖Ͳ_Ϻ𝐹‖_H𝜚 (Ϻ) ≤ c ‖𝐹‖H𝑟(U(Ϻ)).

Conversely, there is also a bounded extension operator EU_Ϻ ∶ H𝜚(Ϻ) → H𝑟(U(Ϻ)),

so for any 𝑔 ∈ H𝜚(Ϻ)

∣∣EU_Ϻ𝑔∣∣

H𝑟(U(Ϻ))≤ c ∣∣𝑔∣∣H𝜚(Ϻ).

Again, this extension operator is independent of 𝜚 and thus universally applicable for any 𝜚 > 0 and 𝑟 = 𝜚 + ĸ/2. All statements remain valid if one replaces U(Ϻ) by

Ϻ and Ϻ by some ESM Γ ∈ 𝕄ℓ_bd(Ϻ) for 0 < ℓ < 𝑘.

The proof of this theorem for ESMs, given in the appendix by deduction from results on linear subspaces, yields in particular the following additional result:

2.33 Corollary — Chart Trace Theorem —

In the setting of the last theorem it holds for any pair (ψ, ω) from the inverse atlas

𝔸_Ϻ∈ ℿ(Ϻ), corresponding (Ψ, Ω) ∈ 𝕌_Ϻ∈ ℿex

N(Ϻ) and 𝐹 ∈ H

𝑟_{(U(Ϻ)) that} ‖Ͳω(𝐹 ∘ Ψ)‖H𝜚(ω) ≤ c ‖𝐹‖H𝑟(Ψ(Ω))≤ c ‖𝐹‖H𝑟(U(Ϻ)).

Conversely, there is a bounded extension operator EU_ω ∶ H𝜚(ω) → H𝑟(U(Ϻ)) such

that for any 𝑔 such that 𝑔 ∘ ψ ∈ H𝜚(ω)

∣∣EUω𝑔∣∣_H𝑟

(U(Ϻ))≤ c ∣∣(𝑔 ∘ ψ)∣∣H𝜚(ω).

2.34 Remark: (1) All results on traces also hold for any other Lipschitz neigh-

the extension results for any Lipschitz neighbourhood of an ESM that is itself con- tained in such a tubular neighbourhood. Both come directly by restriction.

(2) Note that we cannot say anything about traces when 𝜚 = 0. There, the respec- tive results will in fact fail; and while there are other conditions under which one can indeed achieve a trace (cf. [88, Cor. 3.17 etc.]), these conditions will no longer fit into our theory as they apply to 𝑝 ≤ 1 only, so we omit them here.

Nonetheless, there is at least one specific situation where we have some kind of result when taking traces even in case 𝜚 = 0: Namely if we start from Ϻ in the right manner. Then we can give a result on traces that is concerned with extensions based on foliations. Its main purpose is to point out that the trace of a function extended from an ESM into a suitable neighbourhood by the normal (or any other foliation-based) extension is the function we started with:

2.35 Theorem — Foliation Trace Theorem —

Let Ϻ ∈ 𝕄𝑘_bd(ℝ𝑑_{) have the finite inverse atlas 𝔸}

Ϻ = {(ψ𝑖, ω𝑖)}𝑛𝑖=1 ∈ ℿ(Ϻ) and let U(Ϻ) ∈ 𝕃𝕚𝕡_𝑑 be some tubular neighbourhood of Ϻ with fixed extent 𝜚 > 0. If then

𝑓 ∈ W𝑚𝑝(Ϻ) it holds ͲϺEN𝑓 ∈ W 𝑚 𝑝(Ϻ) and ∣∣𝑓 − ͲϺEN𝑓∣∣_W𝑚 𝑝(Ϻ) = 0.

We also have the following result, which gives us an universal extension operator for ESMs that corresponds to the extensions from Euclidean domains to ℝ𝑑:

2.36 Theorem — Manifold Extension Theorem —

Let Ϻ ∈ 𝕄𝑘_bd(ℝ𝑑) be equipped with inverse atlas 𝔸_Ϻ = {(ψ_𝑖, ω_𝑖)}𝑛_𝑖=1 ∈ ℿ(Ϻ). Let

further Ϻ₀∈ 𝕄𝑘

bd(ℝ𝑑) be an open subdomain Ϻ0⋐ Ϻ that is an ESM of dimension 𝑘 in its own right. Then there is an extension operator E ∶ H𝑟(Ϻ0) → H

𝑟

(Ϻ) such

that for any 𝑟 > 0 and any 𝑔 ∈ H𝑟(Ϻ0)

∣∣E𝑔∣∣_H𝑟

(Ϻ) ≤ c ∣∣𝑔∣∣H𝑟(Ϻ0).

This holds also if one replaces Ϻ₀ by an arbitrary parameter space ω with corre-

sponding parameterisation ψ according to the inverse atlas in the sense