Weak Convergence Methods for Nonlinear Partial Differential ...

where
{
(1 − µ)(t 1 − t 2 ), t < µ
z + bψ(t) = µy 1 + (1 − µ)y 2 + b
µ(t 2 − t 1 ), t > µ
{
(1 − µ)(t 1 − t 2 ), t < µ
= µa + µt 1 b + (1 − µ)(a + t 2 b) + b
µ(t 2 − t 1 ), t > µ
{ {
t 1 b
= a +
t 2 b = y 1
y 2 .
Hence,
Now by assumption
w*- lim f(u (ν) ) = µf(y 1 ) + (1 − µ)f(y 2 ).
f(µy 1 + (1 − µ)y 2 ) = f(z) = f(w*- lim u (ν) )
≥ w*- lim f(u (ν) )
= µf(y 1 ) + (1 − µ)f(y 2 ).
As also
Au (ν) = 0 in D ′
(Exercise), the proof is complete.
Examples:
□
1. Suppose Λ = R m (e.g., if A = 0, i.e., no side-conditions). Then l ≥ f(u)
for any sequence satisfying (H 0 ) if and only if f is convex.
2. Λ = 0 (E.g. a i,j,k = a (i1 ,i 2 ),j k
= δ i1 jδ i2 k) leads to the ”compact case”.
3. In the general variational case (see above):
Λ = {a ⊗ b : a ∈ R p , b ∈ R n }.
By Theorem 2.40, a necessary condition for weakly* lower semicontinuity
is that t ↦→ f(A + tB) to be convex ∀B ∈ Λ, i.e., ∀B of rank 1, i.e., ”f is
rank-1-convex”.
Remark 2.41 If f ∈ C 2 (R m ) then convexity in directions of Λ is equivalent to
D 2 f(a)(b, b) ≥ 0 ∀a ∈ R m , b ∈ Λ.
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