Sobolev Space 详细空间理论

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

:



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

(

Namely, in the Sobolev class W

∂

j

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

ess of solution of our variational problem.

theorems and the embedding theorems.

Next, we’ll apply this theory to elliptic boundary value problems.

1

Let us recall some definitions and notation.

An open connected set Ω ⊂ R

is called a domain.

By

Ω we denote the closure of Ω; ∂Ω is the boundary.

If Ω

, ∂Ω} > 0. We use the following

notation:

x = (x

2

, . . . , x

n

) ∈ R

n

, ∂

j

u =

1

2

n

α

∂

...∂x

α

X

j=1

|∂

j

L

Ω

|u(x)|

q

dx



is finite.

L

q

Let u ∈ L

q

(Ω), 1 ≤ q < ∞. We denote

q

dx



Here u(x) is extended by zero on R

\Ω. J

(u; L

continuity of a function u in L

We say that u

−→ 0 for any bounded

Ω

3

Then ω

∈ C

), ω

(x) ≥ 0,

ρ

q

\Ω and consider the convolution ω

ρ

4

2. Properties of mollification

ρ

n

), and

∂

α

(x) =

ρ

(x − y) = 0, y ∈ Ω.

k

q,

kY

k

(

Proof

Case 1

¨

|u

Z

ω

(x − y)



ω

ρ

(x − y)|u(y)|

q

Z

q

=

dy |u(y)|

q

|u(y)|

q

dy

Case 2 : q = ∞. We have

ρ

(x − y)|u(y)|dy