Page 22 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Presentation of the content of the monograph 9

0
Ω
of F (x)=0 in the finite dimensional case, that should satisfy any u ∈ C 2 ¡ ¢
minimizer of (P), namely (we write here the equation in the case N =1)
n
X ∂ £ ¤
(E) f ξ i (x, u, ∇u) = f u (x, u, ∇u) , ∀x ∈ Ω
∂x i
i=1
= ∂f/∂ξ and f u = ∂f/∂u.
where f ξ i i
In thecaseofthe Dirichletintegral
½ Z ¾
1 2
(P) inf I (u)= |∇u (x)| dx : u = u 0 on ∂Ω
2
Ω
the Euler-Lagrange equation reduces to Laplace equation,namely ∆u =0.
2
We immediately note that, in general, finding a C solution of (E) is a difficult
task, unless, perhaps, n =1 or the equation (E) is linear. The next step is to
know if a solution u of (E), called sometimes a stationary point of I,is,in fact,
a minimizer of (P). If (u, ξ) → f (x, u, ξ) is convex for every x ∈ Ω then u is
indeed a minimum of (P); in the above examples this happens for the Dirichlet
integral or the problem of minimal surfaces in nonparametric form. If, however,
(u, ξ) → f (x, u, ξ) is not convex, several criteria, specially in the case n =1,
can be used to determine the nature of the stationary point. Such criteria are
for example, Legendre, Weierstrass, Weierstrass-Erdmann, Jacobi conditions or
the fields theories.
In Chapters 3and 4wewill presentthe direct methods introduced by Hilbert,
Lebesgue and Tonelli. The idea is to break the problem into two pieces: existence
of minimizers in Sobolev spaces and then regularity of the solution. We will start
by establishing, in Chapter 3, the existence of minimizers of (P) in Sobolev spaces
W 1,p (Ω). In Chapter 4 we will see that, sometimes, minimizers of (P) are more
1
regularthaninaSobolevspace they arein C or even in C ,ifthe data Ω, f
∞
and u 0 are sufficiently regular.
We now briefly describe the ideas behind the proof of existence of minimizers
in Sobolev spaces. As for the finite dimensional case we start by considering a
minimizing sequence {u ν } ⊂ W 1,p (Ω), which means that
© 1,p ª
I (u ν ) → inf I (u): u = u 0 on ∂Ω and u ∈ W (Ω) = m,as ν →∞.
The first step consists in showing that the sequence is compact, i.e., that the
sequence converges to an element u ∈ W 1,p (Ω). This, of course, depends on
the topology that we have on W 1,p . The natural one is the one induced by the
norm, that we call strong convergence and that we denote by

u ν → u in W 1,p .