非光滑区域上的椭圆边值问题及齐次化问题

This project proposes to initiate the study of a class of elliptic homogenization problems in domains with non-smooth boundaries. The main focus of this project will be on second order elliptic equations and systems with rapidly oscillating periodic coefficients in non-smooth domains. The primary objective is to gain a better understanding of the boundary regularity properties of solutions by establishing uniform estimates under physically realistic assumptions. We also investigate the sharp ranges of p for which one can solve the boundary value problems with boundary data in L^p for second order elliptic systems, and for higher order elliptic equations. Further, this project consider the Dirichlet, Neumann,regularity problems of Stokes systems in non-smooth domains in the case of high dimension. The proposed research lies at the interface of harmonic analysis and partial differential equations.

本项目主要研究非光滑区域上一族具有高阶震荡系数的二阶椭圆方程和椭圆系统的齐次化问题，主要目标是通过基于实际的物理背景的假设，建立一致估计，从而对方程解的边界正则性有很好的了解和把握。同时我们也考虑高维空间中非光滑区域上使得边值属于L^p空间的二阶椭圆系统和高阶方程唯一可解的p的最佳范围，更进一步的，我们考虑高维情形下的Stokes系统的Dirichlet、Neumann、regularity问题的唯一可解性。本项目的研究属于调和分析与偏微分方程的结合。

我们建立了Lipschitz区域上的Stokes系统的$W^{1,p}$可解性和 $L^p$正则性问题，对任意的 $p>2$, 我们证明了反向H\"older的成立等价于$W^{1,p}$可解性和 $L^p$正则性问题。同时我们也考虑了了一族时空依赖型具有高阶震荡系数的的二阶抛物算子的齐次化问题，我们得到了其一致的内部的$W^{1,p}$估计, H\"older估计, 和Lipschitz估计和边界的$W^{1,p}$估计, H\"older估计。作为一个推论，我们得到了$C^1$区域上的抛物方程组满足Dirichlet边值齐次化问题的$W^{1,p}$可解性。而对于Lipschitz区域上具有周期或者几乎周期系数的椭圆系统的$W^{1,p}$可解性，我们也得到了其成立的$p$的范围。