Page 92 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 3

Direct methods

3.1 Introduction

In this chapter we will study the problem
½ Z ¾
1,p
(P) inf I (u)= f (x, u (x) , ∇u (x)) dx : u ∈ u 0 + W 0 (Ω) = m
Ω
where
n
- Ω ⊂ R is a bounded open set;
n
- f : Ω × R × R −→ R, f = f (x, u, ξ);
1,p 1,p 1,p
- u ∈ u 0 + W (Ω) means that u, u 0 ∈ W (Ω) and u − u 0 ∈ W (Ω)
0 0
(which roughly means that u = u 0 on ∂Ω).
This is the fundamental problem of the calculus of variations. We will show
that the problem (P) has a solution u ∈ u 0 +W 1,p (Ω) provided the two following
0
main hypotheses are satisfied.
(H1) Convexity: ξ → f (x, u, ξ) is convex for every (x, u) ∈ Ω × R;

(H2) Coercivity:there exist p> q ≥ 1 and α 1 > 0, α 2 ,α 3 ∈ R such that
q
n
p
f (x, u, ξ) ≥ α 1 |ξ| + α 2 |u| + α 3 , ∀ (x, u, ξ) ∈ Ω × R × R .
The Dirichlet integral which has as integrand
1 2
f (x, u, ξ)= |ξ|
2
satisfies both hypotheses.

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