Page 119 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

106 Direct methods

2 2×2
Exercise 3.5.1 Show that f (ξ)= (det ξ) ,where ξ ∈ R ,isnot convex.
2
Exercise 3.5.2 Show that if Ω ⊂ R is a bounded open set with Lipschitz bound-
¡
¢
2
ary and if u ∈ v + W 1,p Ω; R ,with p ≥ 2,then
0
ZZ ZZ
det ∇udxdy = det ∇vdxdy .
Ω Ω
¡ ¢
Suggestion: Prove first the result for u, v ∈ C 2 Ω; R 2 with u = v on ∂Ω.
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Exercise 3.5.3 Let Ω ⊂ R be a bounded open set with Lipschitz boundary,
¡
¢
2
u 0 ∈ W 1,p Ω; R ,with p ≥ 2,and
½ ZZ ¾
1,p ¡ 2 ¢
(P) inf I (u)= det ∇u (x) dx : u ∈ u 0 + W Ω; R = m.
0
Ω
Write the Euler-Lagrange equation associated to (P). Is the result totally sur-
prising?
¢
2
Exercise 3.5.4 Let u, v ∈ W 1,p ¡ Ω; R ,with p ≥ 2. Show that there exists
α> 0 (depending only on p) so that
kdet ∇u − det ∇vk L p/2 ≤ α (k∇uk L p + k∇vk L p) k∇u −∇vk L p .
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Exercise 3.5.5 Let Ω ⊂ R be a bounded open set with Lipschitz boundary. We
have seen in Lemma 3.23 that, if p> 2,then

ν
ν
u u in W 1,p ¡ Ω; R 2 ¢ ⇒ det ∇u det ∇u in L p/2 (Ω) .
(i) Show that the result is, in general, false if p =2. To achieve this goal
2
choose, for example, Ω =(0, 1) and
1 ν
ν
u (x, y)= √ (1 − y) (sin νx, cos νx) .
ν
¡ ¢
ν
(ii) Show, using Rellich theorem (Theorem 1.43), that if u ,u ∈ C 2 Ω; R 2
and if p> 4/3 (so in particular for p =2), then
ν
ν
0
u u in W 1,p ¡ Ω; R 2 ¢ ⇒ det ∇u det ∇u in D (Ω) .
(iii) This last result is false if p ≤ 4/3,see Dacorogna-Murat[34].
© ª
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Exercise 3.5.6 Let Ω = x ∈ R : |x| < 1 and u (x)= x/ |x|.
¡ ¢
(i) Show that u ∈ W 1,p Ω; R 2 for every 1 ≤ p< 2 (observe, however, that
0
u/∈ W 1,2 and u/∈ C ).