Weak Convergence Methods for Nonlinear Partial Differential ...

Recall the differential constraints in assumption (H)
( m
)
∑ n∑ ∂u (ν)
Au (ν) j
= a ijk bounded in L 2 (R q )
∂x k
j=1 k=1
i=1,...,q
and the associated linear maps B(ξ) : R m → R q (ξ ∈ R n ) given by
( m
)
∑ n∑
B(ξ)λ = a ijk λ j ξ k .
Also define the sets
j=1 k=1
i=1,...,q
V = {(λ, ξ) ∈ R m × R n : B(ξ)λ = 0} ⊂ R m × R n ,
Λ = {λ ∈ R m : ∃ξ ∈ R n \ {0} : (λ, ξ) ∈ V} ⊂ R m .
Remark 2.39 The less restrictive our constraints that Au (ν) be bounded are, the
bigger V is. If (λ, ξ) ∈ V, λ ≠ 0, ξ ≠ 0, then these constraints do not prevent fast
oscillations with direction Λ in the target space and direction ξ in the original
domain: For a periodic function ψ : R → R let
Then
u (ν) (x) = λψ(νξ · x).
(Au (ν) ) i = ∑ j,k
= ∑ j,k
∂u (ν)
j
a ijk
∂x k
a ijk λ j ψ ′ (νξ · x)νξ k
= ψ ′ (νξ · x)ν(B(ξ)λ) i
= 0.
Examples:
1. No constraints: A = 0. Then Λ = R m .
2.
∂u (ν)
j
∂x k
bounded in L 2 ∀j, k. Then Λ = {0}.
(Note: By Rellich-Kondrachov u (ν) is compact in L 2 . If (H) holds, then
u (ν) → u strongly in L 2 .)
3. If u (ν) : Ω → R m , ∂u(ν) j
∂x k
= 0, k = 2, ..., n, then (H 0 ) holds with
a i,j,k = a i,1,k = δ i+1,k , i = 1, ..., n − 1
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