The Trace and Embedding Theorems for a General Bounded Open Set
Now we show how the primitive versions of the results we have proved (i.e., when U = R n+ )
can be used to deduce analogous results when U is a more general open set. We will
describe now the special properties U must have if this extension of results is to work.
1. Flattening the Boundary
Suppose U is a bounded open set in R n . Then U is said to be regular if:
the boundary ∂U can be covered by open sets O k , k = 1, . . . , M such that for each k,
●
●
●
●
●
pk
qk
pk
pk
∈ C m R n  maps O k onto open set Q
m
n
= p −1
k ∈ C R  maps Q onto O k
+
: O k ∩ U → Q = Q ∩ y n > 0
: O k ∩ ∂U → Q 0 = Q ∩ y n = 0
M
M
k=1
k=0
Then ∂U ⊂ ⋃ O k and we suppose also that Ū ⊂ ⋃ O k , where O 0 denotes an open set in
the interior of U. This property of having a boundary that ”looks like” R n−1 near each of its
points is what will allow us to extend our primitive versions of results to U. We need one
more devise to make this argument work.
We define a partition of unity subordinate to the open covering O k : 0 ≤ k ≤ M. This
is a set of functions a k x ∈ C ∞c R n  such that
b.
supp a k ⊂ O k then a 0 ∈ C ∞c U
a k ≥ 0 on O k
c.
∑ a k x = 1 ∀x ∈ U
a.
M
k=0
Then for f ∈ H m U, we can write
M
fx = a 0 xfx + ∑ a k xfx
k=1
1
=
M
a 0 x f a 0  x + ∑ a k x q #k p #k f a k   x
k=1
where
H m U ∋ f → p # f = fqy ∈ H m Q + 
H m Q +  ∋ g → q # g = gpx ∈ H m U.
Then
f a 0 ∈ H m0 U
p #k f a k  ∈ H m R n+ 
Now define
as
Zf a 0  ∈ H m R n ,
and
with
supp p #k f a k  ⊂ Q + .
A : H m U → H m R n  × H m R n+  M
Af = f a 0 , p #1 f a 1 , . . . , p #M f a M 
and
B : H m R n  × H m R n+  M → H m U
as
M
Bv 0 , . . . , v M  = v 0 a 0 + ∑ a k q #k v k  .
k=1
Then
BAf = f
∀f ∈ H m U.
Evidently, A decomposes f ∈ H m U into M + 1 pieces, one of which lives on O 0 and M
others living on the sets O k ∩ U, 1 ≤ k ≤ M. The mapping B reassembles these pieces into
the original function, f.
Similarly, define
A ′ : H m ∂U → H m R n−1  M and B ′ : H m R n+  M → H m U
by
A ′ f =  p #1 f a 1 , . . . , p #M f a M 
M
B ′ v 1 , . . . , v M  = ∑ a k q #k v k  .
k=1
These two mappings deal only with functions living on the boundary of U so A ′ decomposes
f ∈ H m ∂U into M pieces, living on the sets O k ∩ U, 1 ≤ k ≤ M. The mapping B ′
reassembles these pieces into the original function, f.
2. Basic Extension Lemma
Lemma 2.1 (Basic Extension lemma) For U a bounded, open and regular set in R n , every
u ∈ H m U can be extended to ũ ∈ H m0 V for U ⊂⊂ V.
Proof- Recall the definition, for u ∈ H m R n+ ,
E 1 ux ′ , x n  =
ux ′ , x n 
if
ax n  Eux ′ , x n  if
xn > 0
xn < 0
where
2
ax n  ∈ C ∞ R 1 ,
ax =
1 if x x > 0
= a smooth cutoff function
0 if x n < −1
Then E 1 u ∈ H m R n−1  for all u ∈ H m R n+  where R n−1 = x ′ , x n  : x ′ ∈ R n−1 , − 1 < x n < ∞ .
This modified extension operator smoothly extends the function u ∈ H m R n+  to a
neighborhood of the boundary of R n+ . Now for U ⊂⊂ V we have
H m U − − −A − − > H m R n  × H m R n+  M
↓ id
↓ E1
m
m
n
m
H 0 V < − −B − − − H R  × H R n−1  M
i.e., E 1 extends each function in H m R n+  smoothly to a function in H m R n−1 . Since
R n+ ⊂⊂ R n−1 , applying the mapping B produces a smooth function with support in an open
neighborhood of U.■
3. Trace and Embedding Theorems for a General Open Set
Theorem 3.1 Sobolev Embedding Theorem
For U a bounded, open and regular set in R n , every u ∈ H m U can be identified with
ũ ∈ C k Ū, for m > k + n2 ;
e : H m U ↪ C k Ū, for m > k +
i.e.,
n
2
is a continuous injection.
Proof-
U ⊂⊂ V
H m U − − −A − − > H m R n  × H m R n+  M
↓ id
↓ E1
m
m
n
m
H 0 V < − −B − − − H R  × H R n−1  M
↓Z
m
H R n  − − − − > C k R n  − − − − > C k Ū
where
H m R n  − − − − > C k R n  denotes the continuous injection of theorem 2.1
and
C k R n  − − − − > C k Ū
denotes the restriction from R n to U.
Theorem 3.2 Rellich Embedding Theorem
For U a bounded, open and regular set in R n , the embedding of H m U into H m−1 U is
compact; i.e. any sequence that is bounded in the norm of H m U contains a subsequence
that is convergent in the norm of H m−1 U .
Proof -
U ⊂⊂ V
where
H m U − − −A − − > H m R n  × H m R n+  M
↓ id
↓ E1
m
m
n
m
H 0 V < − −B − − − H R  × H R n−1  M
↓e
H m−1 V − − − − > H m−1 U ,
e : H m0 V − − − > H m−1 V
3
denotes the compact embedding of the corollary to theorem 2.2 and
H m−1 V − − − − > H m−1 U
denotes the restriction from V to U,
Theorem 3.3 The Smooth Approximation Theorem
For U a bounded, open and regular set in R n , C ∞ Ū is dense in H m U.
Proof-
U ⊂⊂ V
C ∞ Ū − − −A − − > H m R n  × H m R n+  M
↓ id
↓ E1
m
m
n
m
C 0 V < − −B − − − H R  × H R n−1  M
↓i
m
H 0 V − − − − > H m U ,
where
C ∞0 V − − > C m0 V − − > H m0 V
is an injection with a dense image since C ∞0 V is dense in both C m0 V and H m0 V so it
follows that C m0 V is dense in H m0 V.
and
H m0 V − − − − > H m U
denotes restriction from V to U■
Theorem 3.4 The Trace Theorem.
For U a bounded, open and regular set in R n ,
T j : H m U − − − > H m−j−1/2 ∂U
0 ≤ j ≤ m − 1,
is a continuous linear surjection and T j u = 0 if and only if u ∈ H m0 U.
Proof
H m U − − −A − − > H m R n  × H m R n+  M
↓0
↓ T j =Primitive Trace Operator
M
m−j−1/2
m−j−1/2
H
0 × H
∂U < − −B − − −
R n−1 
Note
T j Au = T j u a 0 , T j p #1 u a 1 , . . . , T j  p #M u a M 
=
0, ∂ jn p #1 u a 1 , . . . , ∂ jn  p #M u a M 
and
∂ jn p #k u a k  ∈ H m−j−1/2 R n−1  for k = 1, 2, . . . , M
Since the components of A, B and the primitive trace maps are all continuous, the general
trace map is continuous as well (i.e., the composition of continuous mappings is
4
continuous).■
Often we will wish to extend a function defined only on the boundary of a set, into the
interior of the set and be able to say that the extended function belongs to some Sobolev
space on the large set. Here is a theorem that allows us to do this.
Theorem 3.5 Extension from the Boundary to the Interior
Suppose U is a bounded, open and regular set in R n . Then for any f ∈ H m ∂U there exists
̃f ∈ H m U such that T 0 ̃f = f
Proof-
M
H m ∂U − − − −A ′ − − > f #1 , . . . , f #M  ∈ H m R n−1 
↓ K
#
m
′
H U < − − −B − − − Kf 1 , . . . , Kf #M  ∈ H m R n+  M
Here, K denotes the continuous right inverse of the trace operator T 0 .
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