Page 93 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

80 Direct methods

However the minimal surface problem whose integrand is given by
q
2
f (x, u, ξ)= 1+ |ξ|
satisfies (H1) but verifies (H2) only with p =1. Therefore this problem will
require a special treatment (see Chapter 5).
It is interesting to compare the generality of the result with those of the
preceding chapter. The main drawback of the present analysis is that we prove
existence of minima only in Sobolev spaces. In the next chapter we will see that,
under some extra hypotheses, the solution is in fact more regular (for example
1
2
it is C , C or C ).
∞
We now describe the content of the present chapter. In Section 3.2 we con-
sider the model case, namely the Dirichlet integral. Although this is just an
example of the more general theorem obtained in Section 3.3 we will fully discuss
the particular case because of its importance and to make easier the understand-
ing of the method. Recall that the origin of the direct methods goes back to
Hilbert while treating the Dirichlet integral and to Lebesgue and Tonelli. Let
us briefly describe the two main steps in the proof.
1,p
Step 1 (Compactness). Let u ν ∈ u 0 + W 0 (Ω) be a minimizing sequence of
(P), this means that
I (u ν ) → inf {I (u)} = m,as ν →∞.

It will be easy invoking (H2) and Poincaré inequality (cf. Theorem 1.47) to
1,p
obtain that there exists u ∈ u 0 + W (Ω) and a subsequence (still denoted u ν )
0
so that u ν converges weakly to u in W 1,p , i.e.
u ν u in W 1,p ,as ν →∞.
Step 2 (Lower semicontinuity). We will then show that (H1) implies the
(sequential) weak lower semicontinuity of I,namely
u ν u in W 1,p ⇒ lim infI (u ν ) ≥ I (u) .
ν→∞
Since {u ν } was a minimizing sequence we deduce that u is a minimizer of (P).
In Section 3.4 we will derive the Euler-Lagrange equation associated to (P).
Since the solution of (P) is only in a Sobolev space, we will be able to write only
a weak form of this equation.
In Section 3.5 we will say some words about the considerably harder case
N
where the unknown function u is a vector, i.e. u : Ω ⊂ R n −→ R ,with
n, N > 1.
In Section 3.6 we will explain briefly what can be done, in some cases, when
the hypothesis (H1) of convexity fails to hold.