# Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: *L* ^{ p } Estimates

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## Abstract

Let where \({D \subset \mathbb{R}^d (d \geqq 2)}\), is a smooth uniformly convex domain, and which we prove is (generically) sharp for \({d \geqq 4}\). Here for both the oscillating operator and boundary data. For this case, we take \({A_\varepsilon = A(x/ \varepsilon)}\), where

*u*_{ɛ}be a solution to the system$$\rm div(A_\varepsilon(x)\nabla u_\varepsilon(x)) = 0 \quad\text{in}\, D,\quad u_\varepsilon(x) = g(x,x/\varepsilon)\quad\text{on} \,\partial\, D,$$

*g*is 1-periodic in its second variable, and both*A*_{ɛ}and*g*are sufficiently smooth. Our results in this paper are twofold. First we prove*L*^{ p }convergence results for solutions of the above system and for the non oscillating operator \({A_\varepsilon(x) = A(x)}\), with the following convergence rate for all \({1\leqq p < \infty}\)$$\left.\begin{array}{ll}\parallel\,u_\varepsilon - u_0\parallel{L^p(D)} \leqq C_p \left\{\begin{array}{ll}\varepsilon^{1/2p}, \quad\quad\quad\quad d=2, \\(\varepsilon \mid {\rm ln} \varepsilon \mid)^{1/p}, \quad d = 3, \\ \varepsilon^{1/p}, \quad\quad\quad\quad d \geqq 4,\end{array}\right.\end{array}\right.$$

*u*_{0}is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8):1219–1262, 2014), we prove (for certain class of operators and when \({d \geqq 3}\))$$\parallel\,u_\varepsilon - u_0\parallel\,L^p(D) \leqq C_p [\varepsilon ({\rm ln}(1/ \varepsilon))^2 ]^{1/p}$$

*A*is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.## Keywords

Convergence Rate Dirichlet Problem Boundary Data Elliptic System Neumann Problem
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## References

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