Page 40 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Sobolev spaces 27

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Definition 1.31 Let Ω ⊂ R be an open set and 1 ≤ p ≤∞.
p
(i) We let W 1,p (Ω) be the set of functions u : Ω → R, u ∈ L (Ω), whose
p
∈ L (Ω) for every i =1, ..., n.We endow this space
weak partial derivatives u x i
with the following norm
p p 1
kuk W 1,p =(kuk L p + k∇uk L p) p if 1 ≤ p< ∞

kuk W 1,∞ =max {kuk L ∞ , k∇uk L ∞} if p = ∞ .
1
In the case p =2 the space W 1,2 (Ω) is sometimes denoted by H (Ω).
¡ ¢
N
(ii) We define W 1,p Ω; R N to be the set of maps u : Ω → R , u =
¡ 1 N ¢ i 1,p
u , ..., u ,with u ∈ W (Ω) for every i =1, ..., N.
1,p
(iii) If 1 ≤ p< ∞,the set W (Ω) is defined as the closure of C ∞ (Ω)
0 0
functions in W 1,p (Ω). By abuse of language, we will often say, if Ω is bounded,
1,p 1,p
that u ∈ W 0 (Ω) is such that u ∈ W (Ω) and u =0 on ∂Ω.If p =2,the set
1,2 1
W (Ω) is sometimes denoted by H (Ω).
0 0
1,p 1,p
(iv) We will also write u ∈ u 0 + W 0 (Ω) meaning that u, u 0 ∈ W (Ω) and
1,p
u − u 0 ∈ W (Ω).
0
(v) We let W 1,∞ (Ω)= W 1,∞ (Ω) ∩ W 1,1 (Ω).
0 0
(vi) Analogously we define the Sobolev spaces with higher derivatives as fol-
lows. If k> 0 is an integer we let W k,p (Ω) to be the set of functions u : Ω → R,
p
a
whose weak partial derivatives D u ∈ L (Ω), for every multi-index a ∈ A m ,
0 ≤ m ≤ k. The norm will then be
⎧ ! 1
Ã
p
⎪
⎪ P a p
⎪
⎪ kD uk if 1 ≤ p< ∞
⎨ L p
kuk W k,p = 0≤|a|≤k
⎪
⎪
⎪
⎪ a
⎩ max (kD uk L ∞) if p = ∞ .
0≤|a|≤k
(vii) If 1 ≤ p< ∞, W k,p (Ω) is the closure of C ∞ (Ω) in W k,p (Ω) and
0 0
W 0 k,∞ (Ω)= W k,∞ (Ω) ∩ W 0 k,1 (Ω).
If p =2, the spaces W k,2 (Ω) and W 0 k,2 (Ω) are sometimes respectively de-
k
k
noted by H (Ω) and H (Ω).
0