Page 126 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The one dimensional case 113

Proposition 4.1 Let g ∈ C ∞ ([a, b] × R) satisfy

(H2) there exist 2 >q ≥ 1 and α 2 ,α 3 ∈ R such that
q
g (x, u) ≥ α 2 |u| + α 3 , ∀ (x, u) ∈ [a, b] × R .
Let
1 2
f (x, u, ξ)= ξ + g (x, u) .
2
Then there exists u ∈ C ∞ ([a, b]), a minimizer of (P). If, in addition, u →
g (x, u) is convex for every x ∈ [a, b], then the minimizer is unique.

Proof. The existence (and uniqueness, if g is convex) of a solution u ∈
W 1,2 (a, b) follows from Theorem 3.3. We also know from Theorem 3.11 that it
satisfies the weak form of the Euler-Lagrange equation
Z b Z b
0 0 ∞
u v dx = − g u (x, u) vdx, ∀v ∈ C (a, b) . (4.1)
0
a a
To prove further regularity of u,westart by showingthat u ∈ W 2,2 (a, b).This
follows immediately from (4.1) and from the definition of weak derivative. Indeed
2
since u ∈ W 1,2 ,we havethat u ∈ L ∞ and thus g u (x, u) ∈ L ,leading to
¯ ¯
Z b
¯ ¯
¯ 0 0 ¯ ∞
u v dx¯ ≤ kg u (x, u)k L 2 , ∀v ∈ C (a, b) . (4.2)
¯ L 2 kvk 0
¯ a ¯
Theorem 1.36 implies then that u ∈ W 2,2 . We can then integrate by parts (4.1),
bearinginmindthat v (a)= v (b)= 0, and using the fundamental lemma of the
calculus of variations (cf. Theorem 1.24), we deduce that
u (x)= g u (x, u (x)) , a.e. x ∈ (a, b) . (4.3)
00
We are now in a position to start an iteration process. Since u ∈ W 2,2 (a, b)
1
we deduce that (cf. Theorem 1.42) u ∈ C ([a, b]) and hence the function x →
1
g u (x, u (x)) is C ([a, b]), g being C . Returning to (4.3) we deduce that u ∈ C 1
00
∞
3
3
and hence u ∈ C . From there we can infer that x → g u (x, u (x)) is C ,and
3
5
thus from (4.3) we obtain that u ∈ C and hence u ∈ C .Continuing this
00
process we have indeed established that u ∈ C ∞ ([a, b]).
We will now generalize the argument of the proposition and we start with a
lemma.
1
Lemma 4.2 Let f ∈ C ([a, b] × R × R) satisfy (H1), (H2) and (H3’). Then
any minimizer u ∈ W 1,p (a, b) of (P) is in fact in W 1,∞ (a, b) and the Euler-
Lagrange equation holds almost everywhere, i.e.
d
[f ξ (x, u, u )] = f u (x, u, u ) ,a.e. x ∈ (a, b) .
0
0
dx