Page 23 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

10 Introduction

However, it is, in general, not an easy matter to show that the sequence converges
in such a strong topology. It is often better to weaken the notion of convergence
and to consider the so called weak convergence, denoted by . To obtain that

u ν u in W 1,p ,as ν →∞
is much easier and it is enough, for example if p> 1,to show (up to the extraction
of a subsequence) that
ku ν k W 1,p ≤ γ
where γ is a constant independent of ν. Such an estimate follows, for instance,
if we impose a coercivity assumption on the function f of the type
f (x, u, ξ)
lim =+∞, ∀ (x, u) ∈ Ω × R .
|ξ|→∞ |ξ|
2
We observe that the Dirichlet integral, with f (x, u, ξ)= |ξ| /2,satisfies this hy-
pothesis but not the minimal surface in nonparametric form, where f (x, u, ξ)=
q
2
1+ |ξ| .
The second step consists in showing that the functional I is lower semicon-
tinuous with respect to weak convergence, namely
u ν u in W 1,p ⇒ lim infI (u ν ) ≥ I (u) .
ν→∞
We will see that this conclusion is true if
ξ → f (x, u, ξ) is convex, ∀ (x, u) ∈ Ω × R .

Since {u ν } was a minimizing sequence we deduce that u is indeed a minimizer
of (P).

In Chapter 5 we will consider the problem of minimal surfaces. The methods
of Chapter 3 cannot be directly applied. In fact the step of compactness of the
minimizing sequences is much harder to obtain, for reasons that we will detail
in Chapter 5. There are, moreover, difficulties related to the geometrical nature
of the problem; for instance, the type of surfaces that we consider, or the notion
of area. We will present a method due to Douglas and refined by Courant and
Tonelli to deal with this problem. However the techniques are, in essence, direct
methods similar to those of Chapter 3.
n
In Chapter 6 we will discuss the isoperimetric inequality in R . Depending
on the dimension the way of solving the problem is very different. When n =2,
we will present a proof which is essentially the one of Hurwitz and is in the
spirit of the techniques developed in Chapter 2. In higher dimensions the proof
is more geometrical; it will use as a main tool the Brunn-Minkowski theorem.