Page 41 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

28 Preliminaries

p
Remark 1.32 (i) By abuse of notations we will write W 0,p = L .
1
(ii) Roughly speaking, we can say that W 1,p is an extension of C similar to
p
0
that of L as compared to C .
(iii) Note that if Ω is bounded, then
p
C 1 ¡ ¢ 1,∞ (Ω) ⊂ W 1,p (Ω) ⊂ L (Ω)
Ω ⊂ W
6= 6= 6=
for every 1 ≤ p< ∞.
Example 1.33 The following cases are discussed in Exercise 1.4.1.
n −s
(i) Let Ω = {x ∈ R : |x| < 1} and ψ (x)= |x| ,for s> 0. We then have
ψ ∈ L p ⇔ sp < n and ψ ∈ W 1,p ⇔ (s +1) p< n .
© 2 ª s
(ii) Let Ω = x =(x 1 ,x 2 ) ∈ R : |x| < 1/2 and ψ (x)= |log |x|| where
p
0 <s< 1/2. We have that ψ ∈ W 1,2 (Ω), ψ ∈ L (Ω) for every 1 ≤ p< ∞,but
ψ/∈ L ∞ (Ω).
© ª ¡ ¢
2
(iii) Let Ω = x ∈ R : |x| < 1 . We have that u (x)= x/ |x| ∈ W 1,p Ω; R 2
for every 1 ≤ p< 2. Similarly in higher dimensions, namely we will establish
n
that u (x)= x/ |x| ∈ W 1,p (Ω; R ) for every 1 ≤ p< n.
n
Theorem 1.34 Let Ω ⊂ R be open, 1 ≤ p ≤∞ and k ≥ 1 an integer.
(i) W k,p (Ω) equipped with its norm k·k isaBanachspace whichissepa-
k,p
rable if 1 ≤ p< ∞ and reflexive if 1 <p< ∞.
(ii) W 1,2 (Ω) is a Hilbert space when endowed with the following scalar prod-
uct Z Z

hu; vi 1,2 = u (x) v (x) dx + h∇u (x); ∇v (x)i dx .
W
Ω Ω
(iii) The C ∞ (Ω)∩W k,p (Ω) functions are dense in W k,p (Ω) provided 1 ≤ p<
∞.Moreover, if Ω is a bounded domain with Lipschitz boundary (cf. Definition
¡ ¢
1.40), then C ∞ Ω is also dense in W k,p (Ω) provided 1 ≤ p< ∞.
n
n
(iv) W k,p (R )= W k,p (R ), whenever 1 ≤ p< ∞.
0
p
Remark 1.35 (i) Note that as for the case of L the space W k,p is reflexive
only when 1 <p < ∞ and hence W 1,1 is not reflexive; as already said, this is
themainsourceofdifficulties in the minimal surface problem.
(ii) The density result is due to Meyers and Serrin, see Section 7.6 in Gilbarg-
Trudinger [49], Section 5.3 in Evans [43] or Theorem 3.16 in Adams [1].
1,p 1,p n
(iii) In general, we have W 0 (Ω) ⊂ W (Ω),but when Ω = R both coin-
6=
cide (see Corollary 3.19 in Adams [1]).