Weak Convergence Methods for Nonlinear Partial Differential ...

We now prove an important necessary condition for weak lower semicontinuity.
Theorem 2.40 1. If l ≥ f(u) for any sequence satisfying (H 0 ), then f is
convex in the directions of Λ, i.e., t ↦→ f(a + tb) is convex ∀a ∈ R m , b ∈ Λ.
2. If l = f(u) for any sequence satisfying (H 0 ), then f is affine in the directions
of Λ.
In the variational case this is called the “Legendre-Hadamard-” or “ellipticity
condition”.
Proof. Only the first statement is to be shown; the second is an immediate
consequence of the first one.
Let t 1 , t 2 ∈ R, y 1 = a + t 1 b, y 2 = a + t 2 b and µ ∈ (0, 1). b ∈ Λ implies ∃ξ ≠ 0
such that ∑
a ijk b j ξ k = 0 ∀i ∈ {1, ..., q}.
j,k
Let ψ : R → R be 1-periodic with
{
(1 − µ)(t 1 − t 2 ) for 0 ≤ t < µ
ψ(t) =
µ(t 2 − t 1 ) for µ ≤ t < 1.
Define u (ν) ∈ L ∞ (Ω) (Q-periodic with Q = R(0, 1) n , R ∈ SO(n) such that
Re 1 = ξ ) by |ξ|
(
u (ν) (x) = z + bψ ν ξ )
|ξ| · x , z = µy 1 + (1 − µ)y 2 .
Then u (ν) is highly oscillating and converges to
Q
Q
w*- lim u (ν) = z
because
∫ ∫
− u (ν) = z + b− ψ( ξ ∫ 1
|ξ| · x) = z + ψ(t)dt
Similarly,
∫
w*- lim f(u (ν) ) = − f
Q
0
= z + µ(1 − µ)(t 1 − t 2 ) + (1 − µ)µ(t 2 − t 1 ) = z.
( ( )) ξ
z + bψ
|ξ| · x =
∫ 1
0
f(z + bψ(t))dt,
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