Page 113 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

100 Direct methods

(iv) When n, N > 1 the function f should be of the form

f (x, u, ξ)= F (x, u, ξ, adj ξ, adj ξ, ..., adj ξ)
2
s
3
where s =min [n, N] and adj ξ denotes the matrix of all r × r minors of the
r
matrix ξ ∈ R N×n .The hypothesis (H1 vect ) requires then that the function F
be convex for every (x, u) fixed. The hypothesis (H2 vect ) should then hold for
p> max [q, s].
(v) A function f that can be written in terms of a convex function F as in
the theorem is called polyconvex. The theorem is due to Morrey (see also Ball
[7] for important applications of such results to non linear elasticity). We refer
for more details to [31].

Let us now see two examples.
Example 3.21 Let n = N =2, p> 2 and

1 p
f (x, u, ξ)= f (ξ)= |ξ| + h (det ξ)
p
2
where h : R −→ R is non negative and convex (for example h (det ξ)= (det ξ) ).
All hypotheses of the theorem are clearly satisfied. It is also interesting to com-
pute the Euler-Lagrange equations associated. To make them simple consider
¡ ¢
2
1
only the case p =2 and set u = u (x, y)= u (x, y) ,u (x, y) . The system is
then
⎧ 1 £ 2 ¤ £ 2 ¤
0
0
∆u + h (det ∇u) u − h (det ∇u) u =0
⎨ y x x y
£ ¤ £ ¤
⎩ 2 1 1
∆u − h (det ∇u) u + h (det ∇u) u =0 .
0
0
y x x y
Example 3.22 Another important example coming from applications is the fol-
lowing: let n = N =3, p> 3, q ≥ 1 and
p q
f (x, u, ξ)= f (ξ)= α |ξ| + β |adj ξ| + h (det ξ)
2
where h : R −→ R is non negative and convex and α, β > 0.
The key ingredient in the proof of the theorem is the following lemma that
is due to Morrey and Reshetnyak.
2
Lemma 3.23 Let Ω ⊂ R be a bounded open set with Lipschitz boundary, p> 2
and ¡ ¢
ν
ν
ν
u =(ϕ ,ψ ) u =(ϕ, ψ) in W 1,p Ω; R 2 ;
then
ν
det ∇u det ∇u in L p/2 (Ω) .