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arXiv · 1801.09220

The Methods of Layer Potentials for General Elliptic Homogenization Problems in Lipschitz Domains

Qiang Xu·Peihao Zhao·Shulin Zhou
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Abstract

In terms of layer potential methods, this paper is devoted to study the $L^2$ boundary value problems for nonhomogeneous elliptic operators with rapidly oscillating coefficients in a periodic setting. Under a low regularity assumption on the coefficients, we establish the solvability for Dirichlet, regular and Neumann problems in a bounded Lipschitz domain, as well as, the uniform nontangential maximal function estimates and square function estimates. The main difficulty is reflected in two aspects: (i) we can not treat the lower order terms as a compact perturbation to the leading term due to the low regularity assumption; (ii) the nonhomogeneous systems do not possess a scaling-invariant property in general. Although this work may be regarded as a follow-up to C. Kenig and Z. Shen's in \cite{SZW24}, we make an effort to find a clear way of how to handle the nonhomogeneous operators by using the known results of the homogenous ones. Also, we mention that the periodicity condition plays a key role in the scaling-invariant estimates.

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Qiang Xu, Peihao Zhao, Shulin Zhou. 2018-01-28. The Methods of Layer Potentials for General Elliptic Homogenization Problems in Lipschitz Domains. https://arxiv.org/abs/1801.09220

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