Acta Mathematica Sinica, English Series

, Volume 34, Issue 4, pp 612–628 | Cite as

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

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Abstract

We consider a family of second-order elliptic operators {Lε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in ℝn. We are able to show that the uniform W1,p estimate of second order elliptic systems holds for \(\frac{{2n}}{{n + 1}} - \delta < p < \frac{{2n}}{{n - 1}} + \delta \) where δ > 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W1,p estimate is true for \(\frac{3}{2} - \delta < p < 3 + \delta \) if n ≥ 4, similar estimate was extended to convex domains for 1 < p < ∞.

Keywords

Homogenization elliptic non-smooth 

MR(2010) Subject Classification

35B27 35J57 74B05 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouP. R. China

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