Page 45 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

32 Preliminaries

Lemma 1.38 Let u ∈ W 1,p (a, b), 1 ≤ p ≤∞. Then there exists a function
u
e u ∈ C ([a, b]) such that u = e a.e. and
Z x
0
e u (x) − e(y)= u (t) dt, ∀x, y ∈ [a, b] .
u
y
Remark 1.39 (i) As already repeated, we will ignore the difference between u
and e and we will say that if u ∈ W 1,p (a, b) then u ∈ C ([a, b]) and u is the
u
primitive of u , i.e.
0
Z x
u (x) − u (y)= u (t) dt .
0
y
(ii) Lemma 1.38 is a particular case of Sobolev imbedding theorem (cf. below).
It gives a non trivial result, in the sense that it is not, a priori, obvious that a
function u ∈ W 1,p (a, b) is continuous. We can therefore say that

1 1,p
C ([a, b]) ⊂ W (a, b) ⊂ C ([a, b]) , 1 ≤ p ≤∞ .

(iii) The inequality (1.12) in the proof of the lemma below shows that if
u ∈ W 1,p (a, b), 1 <p < ∞,then u ∈ C 0,1/p 0 ([a, b]) and hence u is Hölder
0
continuous with exponent 1/p . We have already seen in Remark 1.37 that if
p = ∞,then C 0,1 ([a, b]) and W 1,∞ (a, b) can be identified.

Proof. We divide the proof into two steps.
Step 1.Let c ∈ (a, b) be fixed and define

Z x
0
v (x)= u (t) dt, x ∈ [a, b] . (1.10)
c
We will show that v ∈ C ([a, b]) and

Z b Z b
0 0 ∞
v (x) ϕ (x) dx = − u (x) ϕ (x) dx, ∀ϕ ∈ C (a, b) . (1.11)
0
a a
Indeed we have
b b x
Z Z µZ ¶
0
v (x) ϕ (x) dx = u (t) dt ϕ (x) dx
0
0
a a c
Z c Z x Z b Z x
0
= dx u (t) ϕ (x) dt + dx u (t) ϕ (x) dt
0
0
0
a c c c