Page 105 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

92 Direct methods

(iii) for every R> 0,there exists β = β (R) such that

³ ´
p p−1
|f u (x, u, ξ)| ≤ β (1 + |ξ| ) and |f ξ (x, u, ξ)| ≤ β 1+ |ξ|
for every x ∈ [a, b] and every u, ξ ∈ R N with |u| ≤ R .

3.4 Euler-Lagrange equations

We now derive the Euler-Lagrange equation associated to (P). The way of pro-
ceeding is identical to that of Section 2.2, but we have to be more careful. Indeed
2
we assumed there that the minimizer u was C , while here we only know that
it is in the Sobolev space W 1,p .

n
Theorem 3.11 Let Ω ⊂ R be a bounded open set with Lipschitz boundary. Let
¡ ¢
n
p ≥ 1 and f ∈ C 1 Ω × R × R , f = f (x, u, ξ), satisfy
(H3) there exists β ≥ 0 so that for every (x, u, ξ) ∈ Ω × R × R n
³ ´
p−1 p−1
|f u (x, u, ξ)| , |f ξ (x, u, ξ)| ≤ β 1+ |u| + |ξ|
¡ ¢
= ∂f/∂ξ and f u = ∂f/∂u.
where f ξ = f ξ 1 , ..., f ξ n , f ξ i i
Let u ∈ u 0 + W 1,p (Ω) be a minimizer of
0
½ Z ¾
(P) inf I (u)= f (x, u (x) , ∇u (x)) dx : u ∈ u 0 + W 0 1,p (Ω) = m
Ω
where u 0 ∈ W 1,p (Ω),then u satisfies the weak form of the Euler-Lagrange equa-
tion
Z
1,p
(E w ) [f u (x, u, ∇u) ϕ + hf ξ (x, u, ∇u); ∇ϕi] dx =0, ∀ϕ ∈ W 0 (Ω) .
Ω
Ω then u satisfies the Euler-
Moreover if f ∈ C 2 ¡ Ω × R × R n ¢ and u ∈ C 2 ¡ ¢
Lagrange equation
n
X ∂ £ ¤
(E) f ξ i (x, u, ∇u) = f u (x, u, ∇u) , ∀x ∈ Ω .
∂x i
i=1
Conversely if (u, ξ) → f (x, u, ξ) is convex for every x ∈ Ω and if u is a
solution of either (E w ) or (E) then it is a minimizer of (P).