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Archive for Rational Mechanics and Analysis

, Volume 215, Issue 1, pp 65–87 | Cite as

Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L p Estimates

  • Hayk Aleksanyan
  • Henrik Shahgholian
  • Per Sjölin
Article

Abstract

Let u ɛ be a solution to the system
$$\rm div(A_\varepsilon(x)\nabla u_\varepsilon(x)) = 0 \quad\text{in}\, D,\quad u_\varepsilon(x) = g(x,x/\varepsilon)\quad\text{on} \,\partial\, D,$$
where \({D \subset \mathbb{R}^d (d \geqq 2)}\), is a smooth uniformly convex domain, and 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 \({1\leqq p < \infty}\)
$$\left.\begin{array}{ll}\parallel\,u_\varepsilon - u_0\parallel{L^p(D)} \leqq C_p \left\{\begin{array}{ll}\varepsilon^{1/2p}, \quad\quad\quad\quad d=2, \\(\varepsilon \mid {\rm ln} \varepsilon \mid)^{1/p}, \quad d = 3, \\ \varepsilon^{1/p}, \quad\quad\quad\quad d \geqq 4,\end{array}\right.\end{array}\right.$$
which we prove is (generically) sharp for \({d \geqq 4}\). Here u 0 is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8):1219–1262, 2014), we prove (for certain class of operators and when \({d \geqq 3}\))
$$\parallel\,u_\varepsilon - u_0\parallel\,L^p(D) \leqq C_p [\varepsilon ({\rm ln}(1/ \varepsilon))^2 ]^{1/p}$$
for both the oscillating operator and boundary data. For this case, we take \({A_\varepsilon = A(x/ \varepsilon)}\), where 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Hayk Aleksanyan
    • 1
  • Henrik Shahgholian
    • 2
  • Per Sjölin
    • 2
  1. 1.School of MathematicsThe University of EdinburghEdinburghUK
  2. 2.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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