Page 16 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Model problem and some examples 3

0.2 Model problem and some examples

We now describe in more detail the problems that we will consider. The model
case takes the following form

½ Z ¾
(P) inf I (u)= f (x, u (x) , ∇u (x)) dx : u ∈ X = m.
Ω

This means that we want to minimize the integral, I (u), among all functions
u ∈ X (and we call m the minimal value that can take such an integral), where

n
- Ω ⊂ R , n ≥ 1, is a bounded open set, a point in Ω will be denoted by
x =(x 1 , ..., x n );
¡ ¢
N
1
- u : Ω → R , N ≥ 1, u = u , ..., u N , and hence
µ j ¶ 1≤j≤N
∂u N×n
∇u = ∈ R ;
∂x i
1≤i≤n
N
- f : Ω × R × R N×n −→ R, f = f (x, u, ξ), is continuous;
- X is the space of admissible functions (for example, u ∈ C 1 ¡ ¢
Ω with u = u 0
on ∂Ω).
We will be concerned with finding a minimizer u ∈ X of (P), meaning that
I (u) ≤ I (u) , ∀u ∈ X.

Many problems coming from analysis, geometry or applied mathematics (in
physics, economics or biology) can be formulated as above. Many other prob-
lems, even though not entering in this framework, can be solved by the very
same techniques.
We now give several classical examples.

Example: Fermat principle.We want to find the trajectory that should
follow a light ray in a medium with non constant refraction index. We can
formulate the problem in the above formalism. We have n = N =1,

q
f (x, u, ξ)= g (x, u) 1+ ξ 2
and
( )
b
Z
(P) inf I (u)= f (x, u (x) ,u (x)) dx : u (a)= α, u (b)= β = m.
0
a