Abstract:
Let $\Omega$ be a Lipschitz domain, and let
$\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$ be a
strongly elliptic operator on $\Omega$ with fairly general boundary conditions,
including, in particular, the Dirichlet and Neumann boundary conditions, as
well as mixed ones.
We suppose that the parameter $\varepsilon$ is small and the function $A$ in
the operator $\mathcal A^\varepsilon$ is Lipschitz in the first variable and
periodic in the second, so its coefficients are locally periodic and rapidly
oscillating.
It is a classical result in homogenization theory that the resolvent $(\mathcal A^\varepsilon-\mu)^{-1}$ converges (in a certain sense)
as $\varepsilon\to0$.
We are interested in approximations for $(\mathcal A^\varepsilon-\mu)^{-1}$
and $\nabla(\mathcal A^\varepsilon-\mu)^{-1}$ in the operator norm on $L_p$
for a suitable $p$.
The rates of the approximations depend on regularity of the effective
operator $\mathcal A^0$.
We prove that if $(\mathcal A^0-\mu)^{-1}$ is continuous from $L_p$ to
the Besov space $B_{p,\infty}^{1+s}$ with $0<s\le1$, then the rates are,
respectively, $\varepsilon^s$ and $\varepsilon^{s/p}$.