Page 35 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

22 Preliminaries

Using Hölder inequality, (1.1) and (1.3) for the first term in the right hand side
of the inequality, we obtain
¯ ¯
¯Z ¯
1 ¯ Z a i ¯
¯ ¯ I P ¯ ¯
¯ u ν (x) ϕ (x) dx ≤ ku ν k L p kϕ − hk L p 0 + |α i | ¯ u ν (x) dx¯
¯
¯ ¯
0 i=1 ¯ a i−1 ¯
(1.4)
¯ ¯
Z
I ¯ a i ¯
P ¯ ¯
≤ kuk L p + |α i | ¯ u ν (x) dx¯ .
i=1 ¯ a i−1 ¯
To conclude we still have to evaluate
Z Z Z
a i a i 1 νa i
u ν (x) dx = u (νx) dx = u (y) dy
ν
a i−1 a i−1 νa i−1
( )
Z [νa i−1 ]+1 Z [νa i ] Z νa i
1
= udy + udy + udy
ν [νa i−1 ]+1 [νa i ]
νa i−1
where [a] stands for the integer part of a ≥ 0. Wenow usethe periodicityof
u in the second term, this is legal since [νa i ] − ([νa i−1 ]+1) is an integer, we
therefore find that
¯ ¯
Z Z 1 ¯Z 1 ¯
¯ a i ¯ 2 ¯
¯ ¯ [νa i ] − [νa i−1 − 1] ¯
u ν (x) dx¯ ≤ |u| dy + ¯ udy .
¯
¯ ¯ ¯
¯ ¯ ν 0 ν 0
a i−1
R 1
Since u = u =0, we have, using the above inequality, and returning to (1.4)
0
¯Z 1 ¯ I
¯ ¯ 2 X
¯ ¯ p + |α i | .
u ν ϕdx ≤ kuk kuk 1
¯ ¯ L ν L
0
i=1
Let ν →∞, we hence obtain
1
¯Z ¯
¯ ¯
0 ≤ lim sup ¯ u ν ϕdx ≤ kuk L p .
¯
¯ ¯
ν→∞ 0
Since is arbitrary, we immediately have (1.2) and thus the result.
We conclude the present Section with a result that will be used on several
occasions when deriving the Euler-Lagrange equation associated to the problems
of the calculus of variations. We start with a definition.
n
Definition 1.23 Let Ω ⊂ R be an open set and 1 ≤ p ≤∞. We say that
p p
0
0
u ∈ L (Ω) if u ∈ L (Ω ) for every open set Ω compactly contained in Ω (i.e.
loc
Ω ⊂ Ω and Ω is compact).
0
0