Page 58 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 2

Classical methods

2.1 Introduction

In this chapter we study the model problem
© 1 ª
(P) inf I (u): u ∈ C ([a, b]) ,u (a)= α, u (b)= β
2
where f ∈ C ([a, b] × R × R) and
Z b
I (u)= f (x, u (x) ,u (x)) dx .
0
a
Before describing the results that we will obtain, it might be useful to recall
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the analogy with minimizations in R ,namely
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inf {F (x): x ∈ X ⊂ R } .
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The methods that we call classical consist in finding x ∈ X satisfying F (x)= 0,
and then analyze the higher derivatives of F so as to determine the nature of the
critical point x: absolute minimizer or maximizer, local minimizer or maximizer
or saddle point.
0
In Section 2.2 we derive the Euler-Lagrange equation (analogous to F (x)= 0
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2
in R )thatshouldsatisfy any C ([a, b]) minimizer, u,of(P),
d
(E) [f ξ (x, u (x) , u (x))] = f u (x, u (x) , u (x)) ,x ∈ [a, b]
0
0
dx
where for f = f (x, u, ξ) we let f ξ = ∂f/∂ξ and f u = ∂f/∂u.
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In general (as in the case of R ), the solutions of (E) are not necessarily
minima of (P); they are merely stationary points of I (cf. below for a more
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