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

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains We consider a family of second-order elliptic operators {L ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in ℝn. We are able to show that the uniform W 1,p estimate of second order elliptic systems holds for $$\frac{{2n}}{{n + 1}} - \delta < p < \frac{{2n}}{{n - 1}} + \delta $$ 2 n n + 1 − δ < p < 2 n n − 1 + δ where δ > 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W 1,p estimate is true for $$\frac{3}{2} - \delta < p < 3 + \delta $$ 3 2 − δ < p < 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 < p < ∞. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Sinica, English Series Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2018 by 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
Subject
Mathematics; Mathematics, general
ISSN
1439-8516
eISSN
1439-7617
D.O.I.
10.1007/s10114-017-7229-5
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Acta Mathematica Sinica, English SeriesSpringer Journals

Published: Nov 28, 2017

References

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