Page 49 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

36 Preliminaries

Remark 1.44 (i) Let us examine the theorems when Ω =(a, b) ⊂ R.Only
cases 2 and 3 apply and in fact we have an even better result (cf. Lemma 1.38),
namely
1
C ([a, b]) ⊂ W 1,p (a, b) ⊂ C 0,1/p 0 ([a, b]) ⊂ C ([a, b])
for every p ≥ 1 (hence even when p =1 we have that functions in W 1,1 are
continuous). However, the imbedding is compact only when p> 1.
(ii) In higher dimension, n ≥ 2,the case p = n cannot be improved, in
general. The functions in W 1,n are in general not continuous and not even
bounded (cf. Example 1.33).
n
(iii) If Ω is unbounded, for example Ω = R , wemustbemorecareful, in
particular, the compactness of the imbeddings is lost (see the bibliography for
more details).
1,p 1,p
(iv) If we consider W instead of W then the same imbeddings are valid,
0
but no restriction on the regularity of ∂Ω is anymore required.
(v) Similar imbeddings can be obtained if we replace W 1,p by W k,p .
(vi) Recall that W 1,∞ (Ω),when Ω is bounded and convex, is identified with
C 0,1 ¡ ¢
Ω .
(vii) We now try to summarize the results when n =1. Ifwedenoteby
I =(a, b),wehave, for p ≥ 1,
D (I)= C 0 ∞ (I) ⊂ ·· · ⊂ W 2,p (I) ⊂ C 1 ¡ ¢ 1,p (I)
I ⊂ W
¡ ¢ 2 1
⊂ C I ⊂ L ∞ (I) ⊂ ··· ⊂ L (I) ⊂ L (I)
1
and furthermore C ∞ is dense in L , equipped with its norm.
0
Theorems 1.42 and 1.43 will not be proved; they have been discussed in
the one dimensional case in Lemma 1.38. Concerning the compactness of the
imbedding when n =1, it is a consequence of Ascoli-Arzela theorem (see Exercise
1.4.4 for more details).
Before proceeding further it is important to understand the significance of
Theorem 1.43. We are going to formulate it for sequences, since it is in this
framework that we will use it. The corollary says that if a sequence converges
p
weakly in W 1,p ,it,in fact,converges strongly in L .
n
Corollary 1.45 Let Ω ⊂ R be a bounded open set with Lipschitz boundary and
1 ≤ p< ∞.If
u ν u in W 1,p (Ω)
p
p
(this means that u ν ,u ∈ W 1,p (Ω), u ν u in L and ∇u ν ∇u in L ). Then
p
u ν → u in L (Ω) .
∗
If p = ∞, u ν u in W 1,∞ ,then u ν → u in L .
∞