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Chapter 1: Sobolev Spaces
Introduction
In many problems of mathematical physics and variational calculus it is not
sufficient to deal with the classical solutions of differential equations. It is
necessary to introduce the notion of weak derivatives and to work in the so
called Sobolev spaces.
Let us consider the simplest example — the Dirichlet problem for the Laplace
equation in a bounded domain Ω ⊂ R
n
:
4u = 0, x ∈ Ω
u(x) = ϕ(x), x ∈ ∂Ω ,
(∗)
where ϕ(x) is a given function on the boundary ∂Ω. It is known that the
Laplace equation is the Euler equation for the functional
l(u) =
Z
Ω
n
X
j=1
∂u
∂x
j
2
dx.
We can consider (∗) as a variational problem: to find the minimum of l(u)
on the set of functions satisfying condition u|
∂Ω
= ϕ. It is much easier to
minimize this functional not in C
1
(
Ω), but in a larger class.
Namely, in the Sobolev class W
1
2
(Ω).
W
1
2
(Ω) consists of all functions u ∈ L
2
(Ω), having the weak derivatives
∂
j
u ∈ L
2
(Ω), j = 1, . . . , n. If the boundary ∂Ω is smooth, then th e trace of
u(x) on ∂Ω is well defined and relation u|
∂Ω
= ϕ makes s en se. (This follows
from the so called
”
boundary trace theorem“ for Sobolev spaces.)
If we consider l(u) on W
1
2
(Ω), it is easy to prove the existence and uniquen-
ess of solution of our variational problem.
The function u ∈ W
1
2
(Ω), that gives minimum to l(u) under the condition
u|
∂Ω
= ϕ, is called the weak solution of the Dirichlet problem (∗).
We’ll study the Sobolev spaces, the extension th eorems, the boundary trace
theorems and the embedding theorems.
Next, we’ll apply this theory to elliptic boundary value problems.
1
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§1: Preliminaries
Let us recall some definitions and notation.
Definition
An open connected set Ω ⊂ R
n
is called a domain.
By
Ω we denote the closure of Ω; ∂Ω is the boundary.
Definition
We say that a domain Ω
0
⊂ Ω ⊂ R
n
is a strictly interior subdomain
of Ω and write Ω
0
⊂⊂ Ω, if
Ω
0
⊂ Ω.
If Ω
0
is bounded and Ω
0
⊂⊂ Ω, then dist {Ω
0
, ∂Ω} > 0. We use the following
notation:
x = (x
1
, x
2
, . . . , x
n
) ∈ R
n
, ∂
j
u =
∂u
∂x
j
,
α = (α
1
, α
2
, . . . , α
n
) ∈ Z
n
+
is a multi–index
|α| = α
1
+ α
2
+ . . . + α
n
, ∂
α
u =
∂
|α|
u
∂x
α
1
1
∂x
α
2
2
...∂x
α
n
n
Next, ∇u = (∂
1
u, . . . , ∂
n
u) , |∇u| =
n
X
j=1
|∂
j
u|
2
1/2
Definition
L
q
(Ω), 1 ≤ q < ∞ , is the set of all measurable functions u(x) in Ω
such that the norm
kuk
q,Ω
=
Z
Ω
|u(x)|
q
dx
1/q
is finite.
L
q
(Ω) is a Banach space. We’ll use the following property:
Let u ∈ L
q
(Ω), 1 ≤ q < ∞. We denote
J
ρ
(u; L
q
) = sup
|z|≤ρ
Z
n
|u(x + z) − u(x)|
q
dx
1/q
.
Here u(x) is extended by zero on R
n
\Ω. J
ρ
(u; L
q
) is called the modulus of
continuity of a function u in L
q
(Ω). Then
J
ρ
(u; L
q
) → 0 as ρ → 0.
2
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Definition
L
q,loc
(Ω), 1 ≤ q < ∞, is th e set of all measurable functions u(x) in Ω
such that
R
Ω
0
|u(x)|
p
dx < ∞ for any bounded strictly interior subdo-
main Ω
0
⊂⊂ Ω.
L
q,loc
(Ω) is a topological space (but not
a Banach space).
We say that u
k
k→∞
−→ u in L
q,loc
(Ω), if ku
k
− uk
q,Ω
0
k→∞
−→ 0 for any bounded
Ω
0
⊂⊂ Ω
Definition
L
∞
(Ω) is the set of all bounded measurable functions in Ω; the norm
is defined by
kuk
∞,Ω
= ess sup
x∈Ω
|u(x)|
Definition
C
l
(
Ω) is the Banach space of all functions in Ω such that u(x) and
∂
α
u(x) with |α| ≤ l are uniformly continuous in Ω and the norm
kuk
C
l
(
Ω)
=
X
|α|≤l
sup
x∈Ω
|∂
α
u(x)|
is finite. If l = 0, we denote C
0
(
Ω) = C(Ω).
Remark
If Ω is bounded, then kuk
C
l
(Ω)
< ∞ follows from the u niform continui-
ty of u, ∂
α
u, |α| ≤ l
Definition
C
l
(Ω) is the class of functions in Ω such that u(x) an d ∂
α
u, |α| ≤ l,
are continuous in Ω.
Remark
Even if Ω is bounded, a function u ∈ C
l
(Ω) may be not bounded; it may
grow near the boundary.
Definition
C
∞
0
(Ω) is the class of the functions u(x) in Ω such that
a) u(x) is infinitely smooth, which means that ∂
α
u is uniformly con-
tinuous in Ω, ∀α;
b) u(x) is compactly supported: supp u is a compact subset of Ω.
3
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§2: Mollification of functions
1. Definition of mollification
The procedure of mollification allows us to approximate function u ∈ L
q
(Ω)
by smooth functions.
Let ω(x), x ∈ R
n
, be a function such that
ω ∈ C
∞
0
(R
n
), ω(x) ≥ 0, ω(x) = 0 if |x| ≥ 1, and
Z
ω(x)dx = 1. (1)
For example, we may take
ω(x) =
(
c exp
n
−
1
1−|x|
2
o
if |x| < 1
0 if |x| ≥ 1
where constant c is chosen so th at condition (1) is satisfied.
For ρ > 0 we p ut
ω
ρ
(x) = ρ
−n
ω
x
ρ
, x ∈ R
n
. (2)
Then ω
ρ
∈ C
∞
0
(R
n
), ω
ρ
(x) ≥ 0,
ω
ρ
(x) = 0 if |x| ≥ ρ, (3)
Z
n
ω
ρ
(x)dx = 1. (4)
Definition
w
ρ
is called a mollifier.
Let Ω ⊂ R
n
be a domain, and let u ∈ L
q
(Ω) with some 1 ≤ q ≤ ∞. We
extend u(x) by zero on R
n
\Ω and consider the convolution ω
ρ
∗u =: u
ρ
u
ρ
(x) =
Z
n
ω
ρ
(x −y)u(y)dy. (5)
In fact, the integral is over Ω ∩ {y : |x − y| < ρ}.
Definition
u
ρ
(x) is called a mollification or regularization of u(x).
4
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2. Properties of mollification
1) u
ρ
∈ C
∞
(R
n
), and
∂
α
u
ρ
(x) =
R
n
∂
α
x
ω
ρ
(x −y)u(y)dy.
This follows from ω
ρ
∈ C
∞
.
2) u
ρ
(x) = 0 if dist {x; Ω} ≥ ρ, since ω
ρ
(x − y) = 0, y ∈ Ω.
3) Let u ∈ L
q
(Ω) with some q ∈ [1, ∞]. Then
ku
ρ
k
q,
n
≤ kuk
q,Ω
. (6)
In other words, the operator Y
ρ
: u 7→ u
ρ
is a linear continuous opera-
tor from L
q
(Ω) to L
q
(R
n
) and
kY
ρ
k
L
q
(Ω)→L
q
(
n
)
≤ 1.
Proof
:
Case 1
: 1 < q < ∞.
Let
1
q
+
1
q
0
= 1. By the H
¨
older inequality and (4), we have
|u
ρ
(x)| =
Z
n
ω
ρ
(x −y)
1/q
ω
ρ
(x −y)
1/q
0
u(y)dy
≤
Z
n
ω
ρ
(x − y)
1/q
0
|
{z }
=1
Z
n
ω
ρ
(x − y)|u(y)|
q
dy
1/q
⇒ |u
ρ
(x)|
q
≤
Z
n
ω
ρ
(x − y)|u(y)|
q
dy
By (4), we obtain
Z
n
|u
ρ
(x)|
q
dx ≤
Z
n
dx
Z
n
ω
ρ
(x − y) |u(y)|
q
dy
=
Z
n
dy |u(y)|
q
Z
n
ω
ρ
(x − y)dx
|
{z }
=1
=
Z
n
|u(y)|
q
dy
Case 2 : q = ∞. We have
|u
ρ
(x)| ≤
Z
n
ω
ρ
(x − y)|u(y)|dy
≤ kuk
∞
Z
n
ω
ρ
(x − y)dy
|
{z }
=1
5
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