Regularity theory of elliptic systems in ε-scale flat domains

We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called ε-scale flatness condition, which could be arbitrarily rough below ε-sca...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2021-03, Vol.379, p.107566, Article 107566
Main Author: Zhuge, Jinping
Format: Article
Language:English
Subjects:
Calderón-Zygmund estimate
Large-scale Lipschitz estimate
Oscillating boundary
Periodic homogenization
ε-scale flatness
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Summary:We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called ε-scale flatness condition, which could be arbitrarily rough below ε-scale. This particularly generalizes Kenig and Prange's work in [34] and [35] by a quantitative approach. Our result also provides a mathematical explanation on why the boundary regularity of the solutions of partial differential equations should be physically and experimentally expected even if the surfaces of mediums in real world may be arbitrarily rough at small scales.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2021.107566