Weak Convergence Methods for Nonlinear Partial Differential ...

for suitable ξ j ∈ L 2 . Passing to the limit in the weak form of (3.13) we get
∫ ∫
ξ · ∇ϕ dx = f ϕ dx (3.27)
for every ϕ ∈ W 1,2
0 (Ω).
We define the “corrector functions” ϕ ε k by
Ω
Ω
ϕ ε k (x) = x k + εχ k
( x
ε
for x ∈ R n . From (3.24) we the deduce that for every ψ ∈ W 1,2
0 (Ω) (extended by
0 outside Ω)
∫
n∑ ( x
a ij
ε
Ω i,j=1
∫
=
∫
=
n∑ ( x
a ij
ε
Ω i,j=1
R n i,j=1
)
)
∂ xj ϕ ε k(x) ∂ xi ψ(x) dx
)( ( x
))
δ jk + ∂ j χ k ∂ xi ψ(x) dx
ε
n∑
a ij (y) ( δ jk + ∂ yj χ k (y) ) ∂ i ψ(εy) ε n dy.
With a partition of unity we can write ψ(εy) = ∑ m∈ 1 2 Zn ψ m (z + y), where
supp ψ m ⊂ Q. But then
∫ n∑ ( x
)
a ij ∂ xj ϕ ε k
Ω ε
(x) ∂ x i
ψ(x) dx
i,j=1
= ε ∑ ∫ n∑
n a ij (y) ( δ jk + ∂ yj χ k (y) ) ∂ i ψ m (z + y) dy = 0
m∈ 1 Q
2 Zn i,j=1
by (3.24) and so ϕ ε k
is a weak solution of
n∑
i,j=1
( ( x
) )
∂ xi a ij ∂ xj ϕ ε k = 0. (3.28)
ε
Let ζ ∈ Cc ∞ (Ω). Since u ε solves (3.13), we have
∫
f(x) ζ(x) ϕ ε k (x) dx
Ω
∫
=
n∑ ( x
a ij
ε
Ω i,j=1
)
∂ xi u ε (x) ( ϕ ε k(x) ∂ xj ζ(x) + ζ(x) ∂ xj ϕ ε k(x) ) dx.
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