Page 103 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

90 Direct methods

Using Hölder inequality and (3.4) we find that for u ν ∈ W 1,p (Ω)
1
f u (x, u, ∇u)(u ν − u) , hf ξ (x, u, ∇u); ∇u ν −∇ui ∈ L (Ω) .
We next integrate (3.3) to get
Z
I (u ν ) ≥ I (u)+ f u (x, u, ∇u)(u ν − u) dx
Ω
(3.5)
Z
+ hf ξ (x, u, ∇u); ∇u ν −∇ui dx .
Ω
p
p
Since u ν − u 0 in W 1,p (i.e. u ν − u 0 in L and ∇u ν −∇u 0 in L )and
p
(3.4) holds, we deduce, from the definition of weak convergence in L ,that
Z Z
lim f u (x, u, ∇u)(u ν − u) dx = lim hf ξ (x, u, ∇u); ∇u ν −∇ui dx =0 .
ν→∞ ν→∞
Ω Ω
Therefore returning to (3.5) we have indeed obtained that
lim infI (u ν ) ≥ I (u) .
ν→∞
Step 3. We now combine the two steps. Since {u ν } was a minimizing se-
quence (i.e. I (u ν ) → inf {I (u)} = m) and for such a sequence we have lower
semicontinuity (i.e. lim inf I (u ν ) ≥ I (u)) we deduce that I (u)= m,i.e. u is a
minimizer of (P).
Part 2 (Uniqueness). The proof is almost identical to the one of Theorem
3.1 and Theorem 2.1. Assume that there exist u, v ∈ u 0 + W 1,p (Ω) so that
0
I (u)= I (v)= m
and we prove that this implies u = v.Denote by w =(u + v) /2 and observe
1,p
that w ∈ u 0 + W (Ω).The function (u, ξ) → f (x, u, ξ) being convex, we can
0
infer that w is also a minimizer since
1 1
m ≤ I (w) ≤ I (u)+ I (v)= m,
2 2
which readily implies that
Z ∙ µ ¶¸
1 1 u + v ∇u + ∇v
f (x, u, ∇u)+ f (x, v, ∇v) − f x, , dx =0 .
2 2 2 2
Ω
The convexity of (u, ξ) → f (x, u, ξ) implies that the integrand is non negative,
while the integral is 0. This is possible only if
µ ¶
1 1 u + v ∇u + ∇v
f (x, u, ∇u)+ f (x, v, ∇v) − f x, , =0 a.e. in Ω .
2 2 2 2
We now use the strict convexity of (u, ξ) → f (x, u, ξ) to obtain that u = v and
∇u = ∇v a.e. in Ω, which implies the desired uniqueness, namely u = v a.e. in
Ω.