Page 114 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The vectorial case 101

Remark 3.24 (i) At first glance the result is a little surprising. Indeed we have
ν
seen in Chapter 1 (in particular Exercise 1.3.3) that if two sequences, say (ϕ ) x
ν
and (ψ ) , converge weakly respectively to ϕ and ψ , then, in general, their
y ν x y
ν
product (ϕ ) (ψ ) does not converge weakly to ϕ ψ . Writing
x y x y
ν
ν
ν
ν
ν
det ∇u =(ϕ ) (ψ ) − (ϕ ) (ψ )
x y y x
ν
ν
ν
ν
we see that both terms (ϕ ) (ψ ) and (ϕ ) (ψ ) do not, in general, converge
x y y x
weakly to ϕ ψ and ϕ ψ but, according to the lemma, their difference, which
x
y
x
y
ν
is det ∇u , converges weakly to their difference, namely det ∇u.We therefore
have a non linear function, the determinant, that has the property to be weakly
continuous. This is a very rare event (see for more details [30] or Theorem 4.2.6
in [31]).
(ii) From Hölder inequality we see that, whenever p ≥ 2 and u ∈ W 1,p then
det ∇u ∈ L p/2 .
(iii) The lemma is false if 1 ≤ p ≤ 2 but remains partially true if p> 4/3;
this will be seen from the proof and from Exercise 3.5.5.
(iv) The lemma generalizes to the case where n, N > 1 and we obtain that
any minor has this property (for example when n = N =3,then any 2 × 2
minor and the determinant are weakly continuous). Moreover they are the only
non linear functions which have the property of weak continuity.
¡ p/2 ¢ 0 p/(p−2)
Proof. We have to show that for every v ∈ L = L
ZZ ZZ
ν
lim det ∇u (x, y) v (x, y) dxdy = det ∇u (x, y) v (x, y) dxdy . (3.8)
ν→∞
Ω Ω
The proof will be divided into three steps. Only the first one carries the impor-
tant information, namely that the determinant has a divergence structure; the
two last steps are more technical. We also draw the attention on a technical
fact about the exponent p.The first step can also be proved if p> 4/3 (cf. also
Exercise 3.5.5). The second, in fact, requires that p ≥ 2 and only the last one
fully uses the strict inequality p> 2. However, in order not to burden the proof
too much, we will always assume that p> 2.
Step 1. We first prove (3.8) under the further hypotheses that v ∈ C 0 ∞ (Ω)
¡ ¢
2
ν
and u ,u ∈ C 2 Ω; R .
We start by proving a preliminary result. If we let v ∈ C 0 ∞ (Ω) and w ∈
¢
2
C 2 ¡ Ω; R , w =(ϕ, ψ), we always have
ZZ ZZ
£ ¤
det ∇wv dxdy = − ϕψ v x − ϕψ v y dxdy . (3.9)
y x
Ω Ω

2

Indeed, using the fact that ϕ, ψ ∈ C , we obtain

¡ ¢
det ∇w = ϕ ψ − ϕ ψ = ϕψ − (ϕψ ) (3.10)
x y y x y x x y