Page 60 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Euler-Lagrange equation 47

Moreover the classical methods suffer of two main drawbacks. The first one
2
1
is that they assume, implicitly, that the solutions of (P) are regular (C , C or
1
sometimes piecewise C ); this is, in general, difficult (or even often false in the
case of (HJ)) to prove. However the main drawback is that they rely on the fact
that we can solve either of the equations (E), (H) or (HJ), which is, usually, not
the case. The main interest in the classical methods is, when they can be carried
completely, that we have an essentially explicit solution. The advantage of the
direct methods presented in the next two chapters is that they do not assume
any solvability of such equations.
We recall, once more, that our presentation is only a brief one and we have
omitted several important classical conditions such as Legendre, Weierstrass,
Weierstrass-Erdmann or Jacobi conditions. The fields theories as well as all the
sufficient conditions for the existence of local minima have only been, very briefly,
presented. We refer for more developments to the following books: Akhiezer [2],
Bliss [12], Bolza [13], Buttazzo-Giaquinta-Hildebrandt [17], Carathéodory [19],
Cesari [20], Courant [25], Courant-Hilbert [26], Gelfand-Fomin [46], Giaquinta-
Hildebrandt [48], Hestenes [56], Pars [82], Rund [90], Troutman [95] or Weinstock
[97].

2.2 Euler-Lagrange equation

The main result of this chapter is

2
Theorem 2.1 Let f ∈ C ([a, b] × R × R), f = f (x, u, ξ),and
( )
b
Z
0
(P) inf I (u)= f (x, u (x) ,u (x)) dx = m
u∈X
a
© 1 ª
where X = u ∈ C ([a, b]) : u (a)= α, u (b)= β .
2
Part 1. If (P) admits a minimizer u ∈ X ∩ C ([a, b]), then necessarily
d
0
(E) [f ξ (x, u (x) , u (x))] = f u (x, u (x) , u (x)) ,x ∈ (a, b)
0
dx
or in other words
0
0
0
f ξξ (x, u (x) , u (x)) u (x)+ f uξ (x, u (x) , u (x)) u (x)
00
0
+f xξ (x, u (x) , u (x)) = f u (x, u (x) , u (x))
0
2
2
2
where we denote by f ξ = ∂f/∂ξ, f u = ∂f/∂u, f ξξ = ∂ f/∂ξ , f xξ = ∂ f/∂x∂ξ
2
and f uξ = ∂ f/∂u∂ξ.