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.
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