Page 118 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The vectorial case 105

sincewecan find a constant γ > 0 so that
1
h ³ ´i p/(p−2)
p/(p−2) (p−2)/2
|h δ (x, det ∇u)| ≤ γ 1+ |det ∇u|
³ ´
p/2
≤ γ 1 1+ |det ∇u| .
Returning to (3.16) and integrating we get

Z
H (u ν ) ≥ H (u)+ h δ (x, det ∇u)(det ∇u ν − det ∇u) dx .
Ω
Since u ν − u 0 in W 1,p , p> 2, we have from Lemma 3.23 that det ∇u ν −
det ∇u 0 in L p/2 which combined with (3.17) and the definition of weak
convergence in L p/2 lead to
Z
lim h δ (x, det ∇u)(det ∇u ν − det ∇u) dx =0 .
ν→∞
Ω
We have therefore obtained that
lim infH (u ν ) ≥ H (u)
ν→∞
and the proof is complete.

3.5.1 Exercises
The exercises will focus on several important analytical properties of the deter-
minant. Although we will essentially deal only with the two dimensional case,
most results, when properly adapted, remain valid in the higher dimensional
cases.
We will need in some of the exercises the following definition.

n
Definition 3.25 Let Ω ⊂ R be an open set and u ν ,u ∈ L 1 (Ω). We say that
loc
u ν converges in the sense of distributions to u, and we denote it by u ν u in
D (Ω),if
0
Z Z
lim u ν ϕdx = uϕ dx, ∀ϕ ∈ C 0 ∞ (Ω) .
ν→∞
Ω Ω
Remark 3.26 (i) If Ω is bounded we then have the following relations
1
∗
0
u ν u in L ∞ ⇒ u ν u in L ⇒ u ν u in D .
(ii) The definition can be generalized to u ν and u that are not necessarily in
L 1 loc (Ω), but are merely what is known as “distributions”, cf. Exercise 3.5.6.