Page 43 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

30 Preliminaries

(i) ⇒ (ii). This follows from Hölder inequality and the fact that u has a weak
derivative; indeed
¯ ¯ ¯ ¯
Z b Z b
¯ ¯ ¯ ¯
¯ 0 ¯ ¯ 0 ¯ 0
¯ u (x) ϕ (x) dx¯ = ¯ u (x) ϕ (x) dx¯ ≤ ku k L p kϕk L p 0 .
a a
¯ ¯ ¯ ¯
(ii) ⇒ (i). Let F be a linear functional defined by
Z b
0
F (ϕ)= hF; ϕi = u (x) ϕ (x) dx, ϕ ∈ C ∞ (a, b) . (1.6)
0
a
Note that, by (ii), it is continuous over C 0 ∞ (a, b).Since C 0 ∞ (a, b) is dense in
p
0
L (a, b) (notethat we used herethe fact that p 6=1 and hence p 6= ∞), we can
0
extend it, by continuity (or appealing to Hahn-Banach theorem), to the whole
0
0
p
p
L (a, b); wehavethereforedefined a continuous linear operator F over L (a, b).
p
From Riesz theorem (Theorem 1.13) we find that there exists v ∈ L (a, b) so
that
Z
b
0
p
F (ϕ)= hF; ϕi = v (x) ϕ (x) dx, ∀ϕ ∈ L (a, b) . (1.7)
a
Combining (1.6) and (1.7) we get
Z b Z b
0
(−v (x)) ϕ (x) dx = − u (x) ϕ (x) dx, ∀ϕ ∈ C ∞ (a, b)
0
a a
p
0
which exactly means that u = −v ∈ L (a, b) and hence u ∈ W 1,p (a, b).
(iii) ⇒ (ii). Let ϕ ∈ C 0 ∞ (a, b) and let ω ⊂ ω ⊂ (a, b) with ω compact and
c
such that supp ϕ ⊂ ω.Let h ∈ R so that |h| <dist (ω, (a, b) ). Wehavethen,
appealing to (iii),
¯ ¯
Z b µZ ¶ 1
¯ ¯ p
p

¯ ¯ |u (x + h) − u (x)| dx
¯ [u (x + h) − u (x)] ϕ (x) dx¯ ≤ kϕk L p 0
¯ a ¯ ω
⎧
⎨ c |h|kϕk L p 0 if 1 <p < ∞
≤ (1.8)
⎩
c |h|kϕk if p = ∞ .
L 1

We know, by hypothesis, that ϕ ≡ 0 on (a, a + h) and (b − h, b) if h > 0 and we

therefore find (letting ϕ ≡ 0 outside (a, b))
b b+h b
Z Z Z
u(x + h) ϕ (x) dx = u(x + h) ϕ (x) dx = u(x) ϕ (x − h) dx . (1.9)
a a+h a