Page 125 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

112 Regularity

The problem of regularity, including the closely related ones concerning reg-
ularity for elliptic partial differential equations, is a difficult one that has at-
tracted many mathematicians. We quote only a few of them: Agmon, Bernstein,
Calderon, De Giorgi, Douglis, E. Hopf, Leray, Liechtenstein, Morrey, Moser,
Nash, Nirenberg, Rado, Schauder, Tonelli, Weyl and Zygmund.
In addition to the books that were mentioned in Chapter 3 one can consult
those by Gilbarg-Trudinger [49] and Ladyzhenskaya-Uraltseva [66].

4.2 The one dimensional case

Let us restate the problem. We consider

( )
Z b
(P) inf I (u)= f (x, u (x) ,u (x)) dx = m
0
u∈X a
© 1,p ª 0
where X = u ∈ W (a, b): u (a)= α, u (b)= β , f ∈ C ([a, b] × R × R), f =
f (x, u, ξ).
We have seen that if f satisfies

(H1) ξ → f (x, u, ξ) is convex for every (x, u) ∈ [a, b] × R;

(H2) there exist p> q ≥ 1 and α 1 > 0, α 2 ,α 3 ∈ R such that
p q
f (x, u, ξ) ≥ α 1 |ξ| + α 2 |u| + α 3 , ∀ (x, u, ξ) ∈ [a, b] × R × R;

then (P) has a solution u ∈ X.
1
If, furthermore, f ∈ C ([a, b] × R × R) and verifies (cf. Remark 3.12)
(H3’) for every R> 0,there exists α 4 = α 4 (R) such that

p
|f u (x, u, ξ)| , |f ξ (x, u, ξ)| ≤ α 4 (1 + |ξ| ) , ∀ (x, u, ξ) ∈ [a, b] × [−R, R] × R .
then any minimizer u ∈ X satisfies the weak form of the Euler-Lagrange equation
Z b
0
0
0
(E w ) [f u (x, u, u ) v + f ξ (x, u, u ) v ] dx =0, ∀v ∈ C ∞ (a, b) .
0
a
We will show that under some strengthening of the hypotheses, we have that if
N
f ∈ C ∞ then u ∈ C . These results are, in part, also valid if u :[a, b] → R ,
∞
for N> 1.
We start with a very elementary result that will illustrate our purpose.