Page 61 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

48 Classical methods

Part 2. Conversely if u satisfies (E) and if (u, ξ) → f (x, u, ξ) is convex for
every x ∈ [a, b] then u is a minimizer of (P).

Part 3. If moreover the function (u, ξ) → f (x, u, ξ) is strictly convex for
every x ∈ [a, b] then the minimizer of (P), if it exists, is unique.

Remark 2.2 (i) One should immediately draw the attention to the fact that this
theorem does not state any existence result.
(ii) As will be seen below it is not always reasonable to expect that the mini-
1
2
mizer of (P) is C ([a, b]) or even C ([a, b]).
(iii) If (u, ξ) → f (x, u, ξ) is not convex (even if ξ → f (x, u, ξ) is convex for
every (x, u) ∈ [a, b] × R) then a solution of (E) is not necessarily an absolute
minimizer of (P). It can be a local minimizer, a local maximizer.... It is often
said that such a solution of (E) is a stationary point of I.
(iv) The theorem easily generalizes, for example (see the exercises below), to
the following cases:
N
- u is a vector, i.e. u :[a, b] → R , N> 1, the Euler-Lagrange equations are
then a system of ordinary differential equations;
- u : Ω ⊂ R n → R, n> 1, the Euler-Lagrange equation is then a single
partial differential equation;
¡ ¢
- f = f x, u, u ,u , ..., u (n) , the Euler-Lagrange equation is then an ordinary
00
0
differential equation of (2n)th order;
- other type of boundary conditions such as u (a)= α, u (b)= β;
0
0
R b
- integral constraints of the form g (x, u (x) ,u (x)) dx =0.
0
a
Proof. Part 1.Since u is a minimizer among all elements of X,wehave
I (u) ≤ I (u + hv)
1
for every h ∈ R and every v ∈ C ([a, b]) with v (a)= v (b)=0.In other words,
1
setting Φ (h)= I (u + hv), wehavethat Φ ∈ C (R) and that Φ (0) ≤ Φ (h) for
every h ∈ R. We therefore deduce that
¯
d ¯
Φ (0) = I (u + hv) ¯ =0
0
dh ¯
h=0
and hence
Z b
[f ξ (x, u (x) , u (x)) v (x)+ f u (x, u (x) , u (x)) v (x)] dx =0 . (2.1)
0
0
0
a
Let us mention that the above integral form is called the weak form of the
Euler-Lagrange equation. Integrating by parts (2.1) we obtain that the following