Page 42 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Sobolev spaces 29

We will now give a simple characterization of W 1,p which will turn out to be
particularly helpful when dealing with regularity problems (Chapter 4).
p
n
Theorem 1.36 Let Ω ⊂ R be open, 1 <p ≤∞ and u ∈ L (Ω). The following
properties are then equivalent.
(i) u ∈ W 1,p (Ω);

(ii) there exists a constant c = c (u, Ω,p) so that

¯Z ¯
¯ ∂ϕ ¯
¯
¯ u (x) (x) dx ≤ c kϕk L p 0 , ∀ϕ ∈ C 0 ∞ (Ω) , ∀i =1, 2, ..., n
¯ ¯
Ω ∂x i
(recalling that 1/p +1/p =1);
0
(iii) there exists a constant c = c (u, Ω,p) so that for every open set ω ⊂
c
n
ω ⊂ Ω,with ω compact, and for every h ∈ R with |h| < dist (ω, Ω ) (where
n
c
Ω = R \ Ω), then
¶ 1
µZ
p
p
|u (x + h) − u (x)| dx ≤ c |h| if 1 <p < ∞
ω
|u (x + h) − u (x)| ≤ c |h| for almost every x ∈ ω if p = ∞ .
Furthermore one can choose c = k∇uk L p in (ii) and (iii).
Remark 1.37 (i)Asa consequenceofthe theorem, it caneasilybeprovedthat
if Ω is bounded and open then
¡ ¢
0,1
C Ω ⊂ W 1,∞ (Ω)
¡ ¢
where C 0,1 Ω has been defined in Section 1.2, and the inclusion is, in general,
strict. If, however, the set Ω is also convex (or sufficiently regular, see Theorem
5.8.4 in Evans [43]), then these two sets coincide (as usual, up to the choice of a
representative in W 1,∞ (Ω)). In other words we can say that the set of Lipschitz
functions over Ω can be identified, if Ω is convex, with the space W 1,∞ (Ω).
(ii) The theorem is false when p =1. Wethenonlyhave(i) ⇒ (ii) ⇔
(iii). The functions satisfying (ii) or (iii) are then called functions of bounded
variations.
Proof. We will prove the theorem only when n =1 and Ω =(a, b).For the
more general case see, for example, Proposition IX.3 in Brézis [14] or Theorem
5.8.3 and 5.8.4 in Evans [43].