Weak Convergence Methods for Nonlinear Partial Differential ...

Although this approach does not lead to a rigorous proof of u ε ⇀ u 0 , the
solution of the equation above, this multi-scale analysis is an important first
step: It provides us with an explicit formula for the homogenized coefficients.
Rigorous analysis:
In the second part of the section we will give a rigorous convergence result.
Suppose u ε is the (unique) weak solution of (3.13). We will now also distinguish
more carefully between the operator L 1 and its formal adjoint L ∗ 1 . (Of course,
L 1 = L ∗ 1 if A is symmetric.) Define χ k = χ k (y), k = 1, . . .,n, to be a weak
solution of the corrector problem
n∑ (
∂ yi aij (y)∂ yj χ k (y) ) =
i,j=1
n∑
∂ yi a ik (y)
i=1
in Q
subject to periodic boundary conditions. More precisely, χ k ∈ Hper 1 (Q) is such
that
∫ n∑
∫ n∑
a ij (y) ∂ yj χ k (y) ∂ yi ψ(y) dy = ∂ yi a ik (y) ψ(y) dy (3.24)
Q i,j=1
Q i=1
for every ψ ∈ Hper(Q). 1 Here the Hilbert space Hper(Q) 1 of Q-periodic H 1 -functions
consists of the restrictions to Q of Q-periodic functions belonging to Hloc 1 (Rn ).
By Fredholm’s alternative, such functions χ k do exist. Also note that χ k is determined
uniquely up to an additive constant. Define the homogenized coefficients
by
∫ n∑
ā ij := a ik (y) (δ kj + ∂ yk χ j (y)) dy.
Q
k=1
Theorem 3.13 u ε converges to u weakly in W 1,2
0 (Ω), where u is the (unique)
weak solution of
−
n∑
∂ xj (ā ij ∂ xi u(x)) = f(x) in Ω, u = 0 on ∂Ω. (3.25)
i,j=1
Proof. As solutions to (3.13), the sequence (u ε ) is bounded in W 1,2
0 (Ω). We may
therefore extract a converging subsequence u ε ⇀ u in W 1,2
0 (Ω). Since the weak
solution of Equation (3.25) is unique, it suffices to show that u solves (3.25).
For a further subsequence (not relabeled) we have
n∑
i=1
( ·
)
a ij ∂ xi u ε ⇀ ξ j (3.26)
ε
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