Page 96 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The model case: Dirichlet integral 83

Therefore returning to the above inequality we have indeed obtained that

lim infI (u ν ) ≥ I (u) .
ν→∞
Step 3. Wenow combinethe twosteps. 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 (D).
1,2
Part 2 (Uniqueness). Assume that there exist u, v ∈ u 0 + W (Ω) so that
0
I (u)= I (v)= m
and let us show that this implies u = v.Denote by w =(u + v) /2 and observe
1,2 2
that w ∈ u 0 + W (Ω). The function ξ → |ξ| being convex, we can infer that
0
w is also a minimizer since
1 1
m ≤ I (w) ≤ I (u)+ I (v)= m,
2 2
which readily implies that
" #
Z ¯ ¯ 2
1 2 1 2 ¯ ∇u + ∇v ¯
|∇u| + |∇v| − ¯ ¯ dx =0 .
2 2 ¯ 2 ¯
Ω
2
Appealing once more to the convexity of ξ → |ξ| , we deduce that the integrand
is non negative, while the integral is 0. This is possible only if
¯ ¯ 2
1 2 1 2 ¯ ∇u + ∇v ¯
|∇u| + |∇v| − ¯ ¯ =0 a.e. in Ω .
2 2 ¯ 2 ¯
2
We now use the strict convexity of ξ → |ξ| to obtain that ∇u = ∇v a.e. in Ω,
which combined with the fact that the two functions agree on the boundary of
1,2
Ω (since u, v ∈ u 0 + W (Ω)) leads to the claimed uniqueness u = v a.e. in Ω.
0
Part 3 (Euler-Lagrange equation). Let us now establish (3.1). Let ∈ R and
1,2 1,2
ϕ ∈ W (Ω) be arbitrary. Note that u + ϕ ∈ u 0 + W (Ω), which combined
0 0
with the fact that u is the minimizer of (D) leads to
Z
1 2
I (u) ≤ I (u + ϕ)= |∇u + ∇ϕ| dx
2
Ω
Z
2
= I (u)+ h∇u; ∇ϕi dx + I (ϕ) .
Ω
The fact that is arbitrary leads immediately to (3.1), which expresses nothing
else than ¯
d ¯
I (u + ϕ) ¯ =0 .
d ¯
=0