In this section we state the existence of the two trace operators which we will use. These trace operators allow us to talk about boundary values ofH1,q (G)-andH2,q(G)-functions. As for ourGthe boundary∂Gis of Lebesgue-measure zero, it makes a priori no sense to talk in anLr-sense about restrictions of such functions to the boundary. The following theorems give us the answer that there is a reasonable way in which we can associate to every such function a
“boundary value”: Although restricting an u∈H1,q(G) to∂G obviously has no sense, any u ∈ H1,q(G) has the property that the restrictions to ∂G of elements of each sequence of C∞(G)-functions converging inH1,q(G)-sense to u also converge in a space Lr(∂G) (which is still to be defined) to a specific function which is only depending on u(and not on the chosen sequence) and which thus in some sense generalizes the notion of boundary value.
In the following, we take a fixed set of charts (∆i, Wi,Φi)i=1,...,N of ∂Gin the sense of∂G∈ C5. Note that the following definitions are at first sight depend-ing highly on the choice of charts. However, bedepend-ing a set of ∂G-measure zero does not depend on this choice and the defined norms on Lr(∂G), H1,r(∂G) are in general different for different choices of charts, but how ever two choices are made, the corresponding norms are equivalent and the corresponding spaces do not depend on this choice, for details we refer to [11], chapitre 2,
§4, chapitre 3, §1. We could also have made these definitions independent of the choice of charts by including the Gram’s determinant-term. However, as this term is bounded from below and from above, we will simply ignore it, which results in different (but still equivalent, which is enough for us as we are only interested in the respective topologies) norms for different choices of charts.
The following definitions introduce the important spaces and state the basic facts which can also be read in [11], chapitre 2,§4, chapitre 3,§1, and in [12], Kapitel 2, 3, too.
Definition 4.1. A subset V ⊂ ∂G is called “of ∂G-measure zero” if and only if for every i= 1, . . . , nthe set (again after an appropriate permutation of variables)
x0 ∈Rn−1 : there is an xn∈R such that x= (x0, xn)∈Wi∩V ⊂∆i is a set of measure zero in Rn−1.
Having now a concept of zero measure, we have again the possibility of saying
“∂G-almost everywhere”.
Definition 4.2. The spaces Lr(∂G), 1< r <∞.
A function f : ∂G → R is said to be in Lr(∂G) if and only if for every i= 1, . . . , N the function
gi : ∆i →R, x0 7→f(x0,Φi(x0)) is in Lr(∆i).
Furthermore,
|f|Lr(∂G):=
N
X
i=1
kgikrr,∆
i
!1r
defines a norm on Lr(∂G).
Definition 4.3. The spaces H1,r(∂G), 1< r <∞.
A function f : ∂G → R is said to be in H1,r(∂G) if and only if for every i= 1, . . . , N the function
gi : ∆i →R, x0 7→f(x0,Φi(x0)) is in H1,r(∆i).
Furthermore,
|f|H1,r(∂G):=
N
X
i=1
kgikr1,r,∆
i
!1r
defines a norm on H1,r(∂G).
Having now introduced the important spaces, we can state the existence of the needed trace operators: Let in the following whenever a r is used in context of a trace operator this r be r = r(q, n) := nq−qn−q if 1 < q < n and r > 1 otherwise. Note that, for our purposes it would suffice, according to the book [2] (see there A6.6, page 265 and A6.10, page 270), to use r =q in every case.
Theorem 4.4. (Compare [12] Satz 2.4.1., page and [11] chapitre 2,th´eor`eme 4.2., page 84)
Let G⊂Rn be a bounded domain with Lipschitz-boundary. Then there exists exactly one linear continuous mapZ1 :H1,q(G)→Lr(∂G)withZ1(u) =u|∂G for all u∈ C∞(G).
Theorem 4.5. (Compare [12] Satz 3.1.3., page and [11] chapitre 2, th´eor`eme 4.11., page 89)
Let G⊂Rn be a bounded domain with Lipschitz-boundary. Then there exists exactly one linear continuous map Z2 : H2,q(G) → H1,r(∂G) with Z2(u) = u|∂G for all u∈ C∞(G).
Remark 4.6. Studying the proofs of Theorems 4.4 and 4.5 given in [12] and [11], one easily sees that the theorems can be modified in the following way:
The linear continuous map Zk fulfills even Zk(u) = u|∂G for all u∈ Ck(G), k = 1,2.
The above defined trace operatorZ1 gives us another characterization of the spaces H01,q(G) andH02,q(G):
Theorem 4.7.
H01,q(G) =
u∈H1,q(G) : Z1(u) = 0
For a proof of this Theorem, see [11], chapitre 2, th´eor`eme 4.10., pages 87, 88 or [12], Satz 2.6.3, pages 40-42.
Theorem 4.8.
H02,q(G) = (
u∈H2,q(G) : Z1(u) = 0 and
n
X
i=1
Z1(∂iu)Ni = 0 )
, where N := (N1, . . . , Nn) denotes the outward unit normal vector.
Proof. This theorem is just a combination of Theorem 4.7, our Theorem 4.10 and Theorem 3.3. For a different proof we refer to [11], chapitre 2, th´eor`eme 4.12., page 90 or [12], Satz 3.2.1, page 45.
The following theorem tells us that the trace-operator behaves very much like a restriction with respect to special kinds of products:
Theorem 4.9. Let s∈H1,q(G) and f ∈ C∞(G). Then Z1(f s)(x) = f(x)Z1(s)(x) for almost every x∈∂G.
Proof. Let (sν)ν∈N ⊂ C∞(G) be a sequence such that ksν −sk1,q →0. This is possible because G is bounded and has continuous boundary, see for ex-ample [9], 1.1.6, Theorem 2, page 14). Then, as f and ∂if, i = 1, . . . , n are bounded in G, we see that f s∈H1,q(G) and (f sν)ν∈N is a Cauchy sequence in H1,q(G) converging to f s. Take a chart (∆i, Wi,Φi) and note that in this chart f sν(x0,Φi(x0)) converges in Lr(∆i) to f Z1(s) as f is bounded and sν
converges in Lr(∆i) to Z1(s), so Z1(f s) must be equal tof Z1(s).
Theorem 4.10. Let s∈H01,q(G)∩H2,q(G). Then we have Z1(∇s)(x) := Z1(∂1s)(x), . . . , Z1(∂ns)(x)
=λ(x)N(x) for almost every x∈∂G with a function λ ∈Lr(∂G).
Proof. The proof is done in three steps:
a) Localization by partition of unity
Let G be covered by finitely many open sets U0, . . . , UN ⊂ Rn such that U0 ⊂⊂ G, ∂G is covered by U1, . . . , UN and for i = 1, . . . , N let Φi ∈ C5(∆i), such that after a permutation of coordinates we have
∂G∩Ui ={(x0,Φi(x0)) : x0 ∈∆i} and G∩Ui = [
x0∈∆i
{x0} ×]Φi(x0),Φi(x0) +εi[ or
G∩Ui = [
x0∈∆i
{x0} ×]Φi(x0)−εi,Φi(x0)[
for real numbers εi >0. We only consider the case G∩Ui = [
x0∈∆i
{x0} ×]Φi(x0),Φi(x0) +εi[ (30) in the following, the other one can be treated in the same manner.
We find a partition of unity Ψi, i = 0, . . . , N of G subordinate to the covering Ui, i= 0, . . . , N.
For j ∈ {1, . . . , n} we find Z1(∂js) =Z1
N
X
l=0
Ψl∂js
!
=Z1
N
X
l=1
Ψl∂js
! ,
as supp Ψ0 ⊂⊂Gand with Theorem 4.9 we see Z1(Ψ0∂js) = 0. More-over, we also see that Z1(∂jΨl·s) = 0 with Theorem 4.9 because s∈H01,q(G). So we get to
Z1(∂js) =Z1
N
X
l=1
∂j(Ψls)
!
and it suffices to show the claim only for functions of the form Ψls.
Moreover, it suffices to show the claim only locally, that is we can take G∩Ul as our newG, which we call G0 and we are searching a function λ ∈ Lr(∂G∩Ul) such that Z1(∂j(Ψls)) = λNj almost everywhere on
∂G∩Ul. In the following we will omit the now fixed indexl.
b) Straightening of a local model
By smoothness of ∂G, we find a C5-diffeomorphism
g :Q:= ∆×]0, ε[→G0, (x0, xn)7→(x0,Φ(x0) +xn),
is the outward unit normal vector to Ginxbecause we are considering the case (30). Taking a functionζ ∈ C∞(G0) we have for the directional
= ∂ζ
∂xi
(g(x0,0)) + ∂ζ
∂xn
(g(x0,0))∂Φ
∂xi
(x0), so
Dtiζ(g(x0,0)) =∂iζ(x˜ 0,0). (31) Taking an approximating sequence ζν in C∞(G0) of Ψs with respect to k·k2,q, we see that ˜ζν := ζν ◦g is an approximating sequence in C5(Q) of ˜s with respect to k·k2,q and we see by equation (31) applied to the approximating sequence and Theorem 4.4 that for almost every p= (x0,Φ(x0))∈∂G∩U we have
Z1(∂is)(x˜ 0,0) =
n
X
j=1
Z1
∂(Ψs)
∂xj
(x0,Φ(x0)) (ti)j(x0,Φ(x0)).
In the following, we will show that Z1(∂i˜s) = 0 and thus we will find that in almost every point p∈∂G0 ∩U it is
n
X
j=1
Z1
∂(Ψs)
∂xj
(p) (ti)j(p) = 0, i= 1, . . . , n−1
and thus in almost every pointp∈∂G0∩U we will then find a λ(p)∈R such that by the definition (∇(Ψs))j(p) :=Z1(∂j(Ψs))(p) forp∈∂G0∩ U we find:
∇(Ψs)(p) =λ(p)N(p) It is easily seen that λ = N1
n∂n(Ψs) is then a measurable function (in the ∂G0∩U-sense) because Nn6= 0 is with the help of Weyers’ helpful function easily to be seen smooth enough, and because of kNk= 1 we also have almost everywhere on ∂G0∩U: |λ(p)|=|∇(Ψs)(p)| and thus λ∈Lr(∂G0∩U).
c) The straight problem:
So we just have to showZ1(∂js)(x˜ 0,0) = 0 forx0 ∈∆. As ˜s ∈H01,q(Q)∩
H2,q(Q) we find a sequence (hν)ν∈N ⊂ C∞(Q) with hν
H2,q(Q)
−−−−→s. With˜ the definition
fν :=hν|∆×{0}
we see that
fν L
r(∆)
−−−→0 and
∂ifν L
r(∆)
−−−→Z1(∂is), i˜ = 1, . . . , n−1
because hν is also an approximating sequence for ˜s with respect to the H1,q(Q)-norm and∂ihν is one in the same norm for ∂is.˜
But we also know that the fν converge in the H1,r(∆)-norm to Z2(˜s) and by choosing subsequences of fν converging almost everywhere on
∆, we conclude Z2(˜s) = 0 and this means that ∂ifν L
r(∆)
−−−→0. Because
∂ihν is an H1,q(Q)-norm approximating sequence for ∂i˜s we also have
∂ifν L
r(∆)
−−−→Z1(∂i˜s) and thusZ1(∂is) = 0.˜