Page 106 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Euler-Lagrange equations 93

Remark 3.12 (i) A more condensed way of writing (E) is

(E) div [f ξ (x, u, ∇u)] = f u (x, u, ∇u) , ∀x ∈ Ω .
(ii) Thehypothesis(H3)isnecessaryfor giving ameaning to (E w ); more
1
precisely for ensuring that f u ϕ, hf ξ ; ∇ϕi ∈ L (Ω). It can be weakened, but only
slightly by the use of Sobolev imbedding theorem (see Exercise 3.4.1).
(iii) Of course any solution of (E) is a solution of (E w ). The converse is
true only if u is sufficiently regular.
(iv) In the statement of the theorem we do not need hypothesis (H1) or (H2)
of Theorem 3.3. Therefore we do not use the convexity of f (naturally for the
converse we need the convexity of f). However we require that a minimizer of
(P)doesexist.
n
(v) The theorem remains valid in the vectorial case, where u : Ω ⊂ R −→
N
R ,with n, N > 1. The Euler-Lagrange equation becomes now a system of
partial differential equations and reads as follows
n
X ∂ h i
(E) f j (x, u, ∇u) = f u j (x, u, ∇u) , ∀x ∈ Ω,j =1, ..., N
ξ
∂x i i
i=1
N
where f : Ω × R × R N×n → R and
µ ¶ 1≤j≤N
³ ´ 1≤j≤N j
¡ 1 N ¢ N j N×n ∂u
u = u , ..., u ∈ R , ξ = ξ ∈ R and ∇u = .
i
1≤i≤n ∂x i
1≤i≤n
(vi) In some casesone can beinterestedinanevenweakerformofthe Euler-
Lagrange equation. More precisely if we choose the test functions ϕ in (E w )to
1,p
be in C ∞ (Ω) instead of in W (Ω) then one can weaken the hypothesis (H3)
0 0
and replace it by
(H3’) there exist p ≥ 1 and β ≥ 0 so that for every (x, u, ξ) ∈ Ω × R × R n
p p
|f u (x, u, ξ)| , |f ξ (x, u, ξ)| ≤ β (1 + |u| + |ξ| ) .
The proof of the theorem remains almost identical. The choice of the space where
the test function ϕ belongs depends on the context. If we want to use the solution,
1,p
u, itself as a test function then we are obliged to choose W (Ω) as the right
0
space (see Section 4.3) while some other times (see Section 4.2) we can actually
limit ourselves to the space C ∞ (Ω).
0
Proof. The proof is divided into four steps.
Step 1 (Preliminary computation). From the observation that
Z 1 d
f (x, u, ξ)= f (x, 0, 0) + [f (x, tu, tξ)] dt, ∀ (x, u, ξ) ∈ Ω × R × R n
0 dt