Page 121 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

108 Direct methods

We have seen in Theorem 3.3 that the existence of minimizers of (P) depend
strongly on the two hypotheses (H1) and (H2). We now brieflydiscuss thecase
where (H1) does not hold, i.e. the function ξ → f (x, u, ξ) is not anymore convex.
We have seen in several examples that in general (P) will have no minimizers.
We present here a way of defining a “generalized” solution of (P). The main
theorem (without proof) is the following.
Theorem 3.28 Let Ω, f, f ∗∗ and u 0 be as above. Let p> 1 and α 1 be such
that
p p n
0 ≤ f (x, u, ξ) ≤ α 1 (1 + |u| + |ξ| ) , ∀ (x, u, ξ) ∈ Ω × R × R .
Finally let
½ Z ¾
1,p
(P) inf I (u)= f ∗∗ (x, u (x) , ∇u (x)) dx : u ∈ u 0 + W (Ω) = m.
0
Ω
Then
(i) m = m;

(ii) for every u ∈ u 0 + W 1,p (Ω),there exists u ν ∈ u 0 + W 1,p (Ω) so that
0 0
u ν u in W 1,p and I (u ν ) → I (u) ,as ν →∞ .
If, in addition, there exist α 2 > 0, α 3 ∈ R such that

p n
f (x, u, ξ) ≥ α 2 |ξ| + α 3 , ∀ (x, u, ξ) ∈ Ω × R × R
then (P) has at least one solution u ∈ u 0 + W 1,p (Ω).
0
Remark 3.29 (i) If f satisfies

p n
f (x, u, ξ) ≥ α 2 |ξ| + α 3 , ∀ (x, u, ξ) ∈ Ω × R × R
p
then its convex envelope f ∗∗ satisfies thesameinequalitysince ξ → α 2 |ξ| +α 3 ≡
h (ξ) is convex and h ≤ f. This observation implies that f ∗∗ verifies (H1)
and (H2) of Theorem 3.3 and therefore the existence of a minimizer of (P) is
guaranteed.
(ii) Theorem 3.28 allows therefore to define u as a generalized solution of
(P), even though (P) may have no minimizer in W 1,p .
(iii) The theorem has been established by L.C. Young, Mac Shane and as
stated by Ekeland (see Theorem 10.3.7 in Ekeland-Témam [41], Corollary 3.13
in Marcellini-Sbordone [71] or [31]). It is false in the vectorial case (see Example
3.31 below). However the author in [29] (see Theorem 5.2.1 in [31]) has shown
that a result in the same spirit can be proved.