Page 88 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Fields theories 75

An elementary passage to the limit leads to Poincaré-Wirtinger inequality

Z 1 Z 1
2
02
u dx ≥ π 2 u dx, ∀u ∈ X.
0 0
For a different proof of a slightly more general form of Poincaré-Wirtinger
inequality see Theorem 6.1.
The way of proceeding, in Theorem 2.21, is, in general, too naive and can be
done only locally; in fact one needs a similar but more subtle theory.

2
Definition 2.24 Let D ⊂ R be a domain. We say that Φ : D → R, Φ =
1
Φ (x, u),isan exact field for f covering D if there exists S ∈ C (D) satisfying
S u (x, u)= f ξ (x, u, Φ (x, u)) = p (x, u)
S x (x, u)= f (x, u, Φ (x, u)) − p (x, u) Φ (x, u)= h (x, u) .
2
Remark 2.25 (i) If f ∈ C , then a necessary condition for Φ to be exact is
that p x = h u . Conversely if D is simply connected and if p x = h u then such an
S exists.
N
(ii) In the case where u :[a, b] → R , N> 1, we have to add to the preceding
,but also (p i ) =(p j ) , for every 1 ≤ i, j ≤
x u j u i
remark, not only that (p i ) = h u i
N.
We startwithanelementaryresultthatisa first justification for defining
such a notion.

2
Proposition 2.26 Let f ∈ C ([a, b] × R × R), f = f (x, u, ξ),and
Z b
I (u)= f (x, u (x) ,u (x)) dx .
0
a
2
1
Let Φ : D → R , Φ = Φ (x, u) be a C exact field for f covering D, [a, b]×R ⊂D.
2
Then any solution u ∈ C ([a, b]) of
0
u (x)= Φ (x, u (x)) (2.20)
solves the Euler-Lagrange associated to the functional I,namely

d
0
0
(E) [f ξ (x, u (x) ,u (x))] = f u (x, u (x) ,u (x)) ,x ∈ [a, b] . (2.21)
dx
Proof. By definition of Φ and using the fact that p = f ξ ,wehave, for any
(x, u) ∈ D,
h u = f u (x, u, Φ)+ f ξ (x, u, Φ) Φ u − p u Φ − pΦ u = f u (x, u, Φ) − p u Φ