Page 36 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

p
L spaces 23

Theorem 1.24 (Fundamental lemma of the calculus of variations). Let
n
Ω ⊂ R be an open set and u ∈ L 1 (Ω) be such that
loc
Z
u (x) ψ (x) dx =0, ∀ψ ∈ C ∞ (Ω) (1.5)
0
Ω
then u =0, almost everywhere in Ω.
Proof. We will show the theorem under the stronger hypothesis that u ∈
2
2
L (Ω) and not only u ∈ L 1 (Ω) (recall that L (Ω) ⊂ L 1 (Ω)); for a proof in
loc loc
the general framework see, for example, Corollary 3.26 in Adams [1] or Lemma
2
IV.2 in Brézis [14] . Let ε> 0.Since u ∈ L (Ω), invoking Theorem 1.13, we
can find ψ ∈ C ∞ (Ω) so that
0
ku − ψk L 2 ≤ ε.
Using (1.5) we deduce that
Z Z
2 2
kuk L 2 = u dx = u (u − ψ) dx .
Ω Ω
Combining the above identity and Hölder inequality, we find
2
kuk L 2 ≤ kuk L 2 ku − ψk L 2 ≤ ε kuk L 2 .
Since ε> 0 is arbitrary we deduce that kuk L 2 =0 and hence the claim.
We next have as a consequence the following result (for a proof see Exercise
1.3.6)
n
Corollary 1.25 Let Ω ⊂ R be an open set and u ∈ L 1 (Ω) be such that
loc
Z Z
u (x) ψ (x) dx =0, ∀ψ ∈ C ∞ (Ω) with ψ (x) dx =0
0
Ω Ω
then u =constant, almost everywhere in Ω.

1.3.1 Exercises
Exercise 1.3.1 (i) Prove Hölder and Minkowski inequalities.
q
p
(ii) Show that if p, q ≥ 1 with pq/ (p + q) ≥ 1, u ∈ L and v ∈ L ,then
uv ∈ L pq/p+q and kuvk L pq/p+q ≤ kuk L p kvk L q .

(iii) Deduce that if Ω is bounded, then
p
q
1
L ∞ (Ω) ⊂ L (Ω) ⊂ L (Ω) ⊂ L (Ω) , 1 ≤ q ≤ p ≤∞ .
Show, by exhibiting an example, that (iii) is false if Ω is unbounded.