Page 21 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

8 Introduction

The second method consists in considering a minimizing sequence {x ν } ⊂ X
so that
F (x ν ) → inf {F (x): x ∈ X} .
We then, with appropriate hypotheses on F, prove that the sequence is compact
in X, meaning that
x ν → x ∈ X,as ν →∞ .
Finally if F is lower semicontinuous, meaning that

lim infF (x ν ) ≥ F (x)
ν→∞
we have indeed shown that x is a minimizer of (P).
We can proceed in a similar manner for problems of the calculus of variations.
The first and second methods are then called, respectively, classical and direct
methods. However, the problem is now considerably harder because we are
working in infinite dimensional spaces.
Let us recall the problem under consideration
½ Z ¾
(P) inf I (u)= f (x, u (x) , ∇u (x)) dx : u ∈ X = m
Ω
where
n
- Ω ⊂ R , n ≥ 1, is a bounded open set, points in Ω are denoted by x =
(x 1 , ..., x n );
³ ´ 1≤j≤N
¡ ¢ ∂u j
1
N
- u : Ω → R , N ≥ 1, u = u , ..., u N and ∇u = ∈ R N×n ;
∂x i
1≤i≤n
N
- f : Ω × R × R N×n −→ R, f = f (x, u, ξ), is continuous;
- X is a space of admissible functions which satisfy u = u 0 on ∂Ω,where u 0
is a given function.
N
Here, contrary to the case of R , we encounter a preliminary problem,
namely: what is the best choice for the space X of admissible functions. A
¡ ¢
natural one seems to be X = C 1 Ω . There are several reasons, which will be
clearer during the course of the book, that indicate that this is not the best
choice. A better one is the Sobolev space W 1,p (Ω), p ≥ 1.We will say that
u ∈ W 1,p (Ω),if u is (weakly) differentiable and if
¸ 1
∙Z
p
p p
kuk 1,p = (|u (x)| + |∇u (x)| ) dx < ∞
W
Ω
The most important properties of these spaces will be recalled in Chapter 1.
In Chapter 2, we will brieflydiscuss the classical methods introduced by
Euler, Hamilton, Hilbert, Jacobi, Lagrange, Legendre, Weierstrass and oth-
ers. The most important tool is the Euler-Lagrange equation,the equivalent