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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 < ∞.

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  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P. R. China

    Jun Geng

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  1. Jun Geng
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Correspondence to Jun Geng.

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Geng, J. Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains. Acta. Math. Sin.-English Ser. 34, 612–628 (2018). https://doi.org/10.1007/s10114-017-7229-5

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  • DOI: https://doi.org/10.1007/s10114-017-7229-5

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