Applications of Fourier analysis in homogenization of Dirichlet problem II. $L^p$ estimates

Let $u_\e$ be a solution to the system $$\mathrm{div}(A_\e(x) \nabla u_{\e}(x))=0 \text{\ in} D, \qquad u_{\e}(x)=g(x,x/\e) \text{\ on}\partial D,$$ where $D \subset \R^d$ ($d \geq 2$), is a smooth uniformly convex domain, and $g$ is 1-periodic in its second variable, and both $A_\e$ and $g$ reasonably smooth. Our results in this paper are two folds. First we prove $L^p$ convergence results for solutions of the above system, for non-oscillating operator, $A_\e(x) =A(x)$, with the following convergence rate for all $1\leq p <\infty$ $$\|u_\e - u_0\|_{L^p(D)} \leq C_p \begin{cases} \e^{1/2p} ,&\text{$d=2$}, (\e |\ln \e |)^{1/p}, &\text{$d = 3$}, \e^{1/p} ,&\text{$d \geq 4$,} \end{cases}$$ which we prove is (generically) sharp for $d\geq 4$. Here $u_0$ is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen \cite{KLS1}, we prove (for certain class of operators and when $d\geq 3$) $$|| u_\e - u_0 ||_{L^p(D)} \leq C_p [ \e (\ln(1/ \e))^2 ]^{1/p}.$$ for both oscillating operator and boundary data. For this case, we take $A_\e=A(x/\e)$, where $A$ is 1-periodic as well. Some further applications of the method to the homogenization of Neumann problem with oscillating boundary data are also considered.