Page 86 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Fields theories 73

Let us recall the problem under consideration

( )
b
Z
(P) inf I (u)= f (x, u (x) ,u (x)) dx = m
0
u∈X
a
© ª
1
where X = u ∈ C ([a, b]) : u (a)= α, u (b)= β . The Euler-Lagrange equa-
tion is
d
0
(E) [f ξ (x, u, u )] = f u (x, u, u ) ,x ∈ (a, b) .
0
dx
We will try to explain the nature of the theory, starting with a particularly
simple case. We have seen in Section 2.2 that a solution of (E) is not, in general,
a minimizer for (P). However (cf. Theorem 2.1) if (u, ξ) → f (x, u, ξ) is convex
for every x ∈ [a, b] then any solution of (E) is necessarily a minimizer of (P). We
first show that we can, sometimes, recover this result under the only assumption
that ξ → f (x, u, ξ) is convex for every (x, u) ∈ [a, b] × R.
3
2
Theorem 2.21 Let f ∈ C ([a, b] × R × R).If there exists Φ ∈ C ([a, b] × R)
with Φ (a, α)= Φ (b, β) such that
(u, ξ) → f (x, u, ξ) is convex for every x ∈ [a, b]
e
where
f (x, u, ξ)= f (x, u, ξ)+ Φ u (x, u) ξ + Φ x (x, u);
e
then any solution u of (E) is a minimizer of (P).

Remark 2.22 We should immediately point out that in order to have (u, ξ) →
f (x, u, ξ) convex for every x ∈ [a, b] we should, at least, have that ξ → f (x, u, ξ)
e
is convex for every (x, u) ∈ [a, b] × R.If (u, ξ) → f (x, u, ξ) is already convex,
then choose Φ ≡ 0 and apply Theorem 2.1.

Proof. Define

ϕ (x, u, ξ)= Φ u (x, u) ξ + Φ x (x, u) .

Observe that the two following identities (the first one uses that Φ (a, α)=
Φ (b, β) and the second one is just straight differentiation)

Z
b
d
[Φ (x, u (x))] dx = Φ (b, β) − Φ (a, α)= 0
dx
a
d £ 0 ¤ 0
ϕ (x, u, u ) = ϕ (x, u, u ) ,x ∈ [a, b]
ξ
u
dx