Page 107 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

94 Direct methods

and from (H3), we find that there exists γ > 0 so that
1
p p n
|f (x, u, ξ)| ≤ γ (1 + |u| + |ξ| ) , ∀ (x, u, ξ) ∈ Ω × R × R . (3.6)
1
In particular we deduce that

|I (u)| < ∞, ∀u ∈ W 1,p (Ω) .

Step 2 (Derivative of I). We now prove that for every u, ϕ ∈ W 1,p (Ω) and
every ∈ R we have
Z
I (u + ϕ) − I (u)
lim = [f u (x, u, ∇u) ϕ + hf ξ (x, u, ∇u); ∇ϕi] dx . (3.7)
→0
Ω
We let
g (x, )= f (x, u (x)+ ϕ (x) , ∇u (x)+ ∇ϕ (x))
so that Z
I (u + ϕ)= g (x, ) dx .
Ω
1
1
Since f ∈ C we have, for almost every x ∈ Ω,that → g (x, ) is C and
therefore there exists θ ∈ [− | | , | |], θ = θ (x), such that
g (x, ) − g (x, 0) = g (x, θ)

where
g (x, θ)= f u (x, u + θϕ, ∇u + θ∇ϕ) ϕ + hf ξ (x, u + θϕ, ∇u + θ∇ϕ); ∇ϕi .

The hypothesis (H3) implies then that we can find γ > 0 so that, for every
2
θ ∈ [−1, 1],
¯ ¯
¯ g (x, ) − g (x, 0) ¯ p p p p
¯ ¯ = |g (x, θ)| ≤ γ (1 + |u| + |ϕ| + |∇u| + |∇ϕ| ) ≡ G (x) .
2

¯ ¯
1
Note that since u, ϕ ∈ W 1,p (Ω),wehave G ∈ L (Ω).
We now observe that, since u, ϕ ∈ W 1,p (Ω), we have from (3.6) that the
1
functions x → g (x, 0) and x → g (x, ) are both in L (Ω).
Summarizing the results we have that
g (x, ) − g (x, 0) 1
∈ L (Ω) ,

¯ ¯
¯ g (x, ) − g (x, 0) ¯ 1
¯ ¯ ≤ G (x) ,with G ∈ L (Ω)

¯ ¯
g (x, ) − g (x, 0)
→ g (x, 0) a.e. in Ω .