Page 98 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

A general existence theorem 85

Remark 3.4 (i) The hypotheses of the theorem are nearly optimal, in the sense
that the weakening of any of them leads to a counterexample to the existence
of minima (cf. below). The only hypothesis that can be slightly weakened is the
continuity of f (see the above mentioned literature).
(ii) The proof will show that uniqueness holds under a slightly weaker con-
dition, namely that (u, ξ) → f (x, u, ξ) is convex and either u → f (x, u, ξ) is
strictly convex or ξ → f (x, u, ξ) is strictly convex.
n
(iii) The theorem remains valid in the vectorial case, where u : Ω ⊂ R −→
N
R ,with n, N > 1. However the hypothesis (H1) is then far from being optimal
(cf. Section 3.5).
(iv) This theorem has a long history and we refer to [31] for details. The
first one that noticed the importance of the convexity of f is Tonelli.

Before proceeding with the proof of the theorem, we discuss several examples,
emphasizing the optimality of the hypotheses.

Example 3.5 (i) The Dirichlet integral considered in the preceding section en-
ters, of course, in the framework of the present theorem; indeed we have that

1 2
f (x, u, ξ)= f (ξ)= |ξ|
2
satisfies all the hypotheses of the theorem with p =2.
(ii) The natural generalization of the preceding example is

1 p
f (x, u, ξ)= |ξ| + g (x, u)
p
where g is continuous and non negative and p> 1.

Example 3.6 The minimal surface problem has an integrand given by

q
2
f (x, u, ξ)= f (ξ)= 1+ |ξ|
that satisfies all the hypotheses of the theorem but (H2), this hypothesis is only
verified with p =1. We will see in Chapter 5 that this failure may lead to
non existence of minima for the corresponding (P). The reason why p =1 is not
allowed is that the corresponding Sobolev space W 1,1 is not reflexive (see Chapter
1).

Example 3.7 This example is of the minimal surface type but easier, it also
shows that all the hypotheses of the theorem are satisfied, except (H2) that is