Page 37 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

24 Preliminaries

Exercise 1.3.2 Establish the results in Example 1.17.

Exercise 1.3.3 (i) Prove that if 1 ≤ p< ∞,then
u ν u in L p ⎫
⎬
1
⇒ u ν v ν uv in L .
v ν → v in L p 0 ⎭
Find an example showing that the result is false if we replace v ν → v in L p 0 by
p
0
v ν v in L .
(ii) Show that
⎫
2
u ν u in L ⎬
2
⇒ u ν → u in L .
2
2
u u in L 1 ⎭
ν
Exercise 1.3.4 (Mollifiers). Let ϕ ∈ C ∞ (R), ϕ ≥ 0, ϕ (x)= 0 if |x| > 1 and
0
R
+∞
ϕ (x) dx =1,for example
−∞
⎧ ½ ¾
1
⎨ c exp 2 if |x| < 1
⎪
⎪
ϕ (x)= x − 1
⎪
⎪
⎩
0 otherwise
R +∞
and c is chosen so that ϕdx =1.Define
−∞
ϕ (x)= νϕ (νx)
ν
Z
+∞
u ν (x)= (ϕ ∗ u)(x)= ϕ (x − y) u (y) dy .
ν ν
−∞
(i) Show that if 1 ≤ p ≤∞ then
ku ν k L p ≤ kuk L p .
p
(ii) Prove that if u ∈ L (R),then u ν ∈ C ∞ (R).
(iii) Establish that if u ∈ C (R),then
u ν → u uniformly on every compact set of R.
p
(iv) Show that if u ∈ L (R) and if 1 ≤ p< ∞,then
p
u ν → u in L (R) .
Exercise 1.3.5 In Step 2 of Theorem 1.22 use approximation by smooth func-
tions instead of by step functions.