Page 186 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 1: Preliminaries 173

(ii) The result follows, since ϕ is C ∞ and

Z
+∞
0
u (x)= ϕ (x − y) u (y) dy .
0
ν ν
−∞
(iii) Let K ⊂ R be a fixed compact. Since u is continuous, we have that for
every > 0,there exists δ = δ ( , K) > 0 so that
|y| ≤ δ ⇒ |u (x − y) − u (x)| ≤ , ∀x ∈ K.
R R
Since ϕ =0 if |x| > 1, ϕ =1, and hence ϕ =1,we find that
ν
Z
+∞
u ν (x) − u (x)= [u (x − y) − u (x)] ϕ (y) dy
ν
−∞
1/ν
Z
= [u (x − y) − u (x)] ϕ (y) dy .
ν
−1/ν
Taking x ∈ K and ν> 1/δ, we deduce that |u ν (x) − u (x)| ≤ , and thus the
claim.
p
(iv) Since u ∈ L (R) and 1 ≤ p< ∞, we deduce (see Theorem 1.13) that for
every > 0,there exists u ∈ C 0 (R) so that
ku − uk p ≤ . (7.1)
L
Define then
Z
+∞
u ν (x)= (ϕ ∗ u)(x)= ϕ (x − y) u (y) dy .
ν ν
−∞
p
Since u − u ∈ L , it follows from (i) that
ku ν − u ν k L p ≤ ku − uk L p ≤ . (7.2)
Moreover, since supp u is compact and ϕ =0 if |x| > 1,we find that there exists
acompact set K so that supp u, supp u ν ⊂ K (for every ν). From (iii) we then
get that ku ν − uk p → 0. Combining (7.1) and (7.2), we deduce that
L
ku ν − uk ≤ ku ν − u ν k
L p L p + ku ν − uk L p + ku − uk L p
≤ 2 + ku ν − uk
L p
which is the claim, since is arbitrary.
Exercise 1.3.5. We adopt the same hypotheses and notations of Theorem 1.22.
Step 1 remains unchanged and we modify Step 2 as follows.