Page 120 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Relaxation theory 107

ν
ν
(ii) Let u (x)= x/ (|x| +1/ν). Show that u u in W 1,p ,for any 1 ≤ p<
2.
(iii) Let δ (0,0) be the Dirac mass at (0, 0), which means
­ ®
δ (0,0) ; ϕ = ϕ (0, 0) , ∀ϕ ∈ C ∞ (Ω) .
0
Prove that
ν
det ∇u πδ (0,0) in D (Ω) .
0
3.6 Relaxation theory

Recall that the problem under consideration is
½ Z ¾
1,p
(P) inf I (u)= f (x, u (x) , ∇u (x)) dx : u ∈ u 0 + W (Ω) = m
0
Ω
where
n
- Ω ⊂ R is a bounded open set with Lipschitz boundary;
n
- f : Ω × R × R −→ R, f = f (x, u, ξ), is continuous, uniformly in u with
respect to ξ;
- u 0 ∈ W 1,p (Ω) with I (u 0 ) < ∞.

Before stating the main theorem, let us recall some facts from Section 1.5.
Remark 3.27 The convex envelope of f, with respect to the variable ξ,willbe
∗∗
denoted by f . It is the largest convex function (with respect to the variable ξ)
which is smaller than f.In other words
g (x, u, ξ) ≤ f ∗∗ (x, u, ξ) ≤ f (x, u, ξ) , ∀ (x, u, ξ) ∈ Ω × R × R n

for every convex function g (ξ → g (x, u, ξ) is convex), g ≤ f. We have two ways
of computing this function.
(i) From the duality theorem (Theorem 1.54) we have, for every (x, u, ξ) ∈
n
Ω × R × R ,
∗ ∗ ∗
f (x, u, ξ )= sup {hξ; ξ i − f (x, u, ξ)}
ξ∈R n
∗
∗
f ∗∗ (x, u, ξ)= sup {hξ; ξ i − f (x, u, ξ )} .
∗
ξ ∈R n
∗
(ii) From Carathéodory theorem (Theorem 1.55) we have, for every (x, u, ξ) ∈
n
Ω × R × R ,
( )
n+1 n+1 n+1
X X X
f ∗∗ (x, u, ξ)= inf λ i f (x, u, ξ ): ξ = λ i ξ ,λ i ≥ 0 and λ i =1 .
i
i
i=1 i=1 i=1