On Homogenization for Piecewise Locally Periodic Operators
Tóm tắt
We discuss homogenization of a strongly elliptic operator
$$\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon_\#)\nabla$$
on a bounded
$$C^{1,1}$$
domain in
$$\mathbb R^d$$
with either Dirichlet or Neumann boundary condition. The function
$$A$$
is piecewise Lipschitz in the first variable and periodic in the second one, and the function
$$\varepsilon_\#$$
is identically equal to
$$\varepsilon_i(\varepsilon)$$
on each piece
$$\Omega_i$$
, with
$$\varepsilon_i(\varepsilon)\to0$$
as
$$\varepsilon\to0$$
. For
$$\mu$$
in a resolvent set, we show that the resolvent
$$(\mathcal A^\varepsilon-\mu)^{-1}$$
converges, as
$$\varepsilon\to0$$
, in the operator norm on
$$L_2(\Omega)^n$$
to the resolvent
$$(\mathcal A^0-\mu)^{-1}$$
of the effective operator at the rate
$$\varepsilon_ {\vee} $$
, where
$$\varepsilon_ {\vee} $$
stands for the largest of
$$\varepsilon_i(\varepsilon)$$
. We also obtain an approximation for the resolvent in the operator norm from
$$L_2(\Omega)^n$$
to
$$H^1(\Omega)^n$$
with error of order
$$\varepsilon_ {\vee} ^{1/2}$$
.
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