Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 29

Introduction to Sobolev Spaces

0, let H m () be the set of all functions u
in L 2 () which possess weak derivatives ∂ α u for all

1.2 Definition. Given an integer m ≥

| α |≤ m . We can define a

scalar product on H m () by
(u, v) m : (∂ α u, ∂ α v) 0

=

| α |≤ m

with the associated norm

(u, u) m =

2

u m : =

m ∂ α u

L 2 () .

( 1 . 4 )

| |≤

α

The corresponding semi-norm

| α |= m ∂ α u

2
L 2 ()

| u | m : = ( 1 . 5 )

is also of interest.

We shall often write H m instead of H m () . Conversely, we will write · m,

instead of · m whenever it is important to distinguish the domain.

The letter H is used in honor of David Hilbert.
H m () is complete with respect to the norm

· m , and is thus a Hilbert
space. We shall make use of the following result which is often used to introduce
the Sobolev spaces without recourse to the concept of weak derivative.

n

1.3 Theorem. Let ⊂ R be an open set with piecewise smooth boundary, and

0 . Then C ∞ () ∩ H m () is dense in H m ().
By Theorem 1.3, H m () is the completion of C ∞ () ∩ H m () , provided
that is bounded. This suggests a corresponding generalization for functions
satisfying zero boundary conditions.

let m ≥

1.4 Definition. We denote the completion of C 0 () w.r.t. the Sobolev norm

· m

by H 0 () .
Obviously, the Hilbert space H 0 () is a closed subspace of H m () . More-
over, H 0 () = L 2 () , and we have the following inclusions:
L 2 () = H 0 () ⊃ H 1 () ⊃ H 2 () ⊃ ···
∪ ∪
H 0 () ⊃ H 0 () ⊃ H 0 () ⊃ ···
The above Sobolev spaces are based on L 2 () and the L 2 -norm. Analogous
Sobolev spaces can be defined for arbitrary L p -norms with p =

2. They are useful
in the study of nonlinear elliptic problems. We denote the spaces analogous to H m
and H 0

and W m,p

by W m,p

, respectively.

0