几乎周期抛物方程组的均匀化和Lipschitz区域上的高阶抛物方程组

This project proposes to initiate the study of almost-periodic homogenization of parabolic systems and $L^p$ solvability of higher order parabolic systems in Lipschitz domains. The main objective is to obtain a better understanding of the boundary regularity properties of solutions via establishing the convergence rates and the a-prior estimates such as interior and boundary Lipschitz estimates, H\"older estimates and $W^{1,p}$ estimates， especially，the primary focus of this project will be on second order parabolic systems subject to Robin or mix boundary conditions with rapidly oscillating periodic or almost-periodic coefficients. We also investigate the sharp ranges of p for which one may solve the $L^p$ boundary value problems for higher order parabolic systems in Lipschitz domains. The proposed research lies at the interface of partial differential equations, compensated compactness method and harmonic analysis as well as functional analysis.

本项目主要研究具有几乎周期系数的抛物方程组的均匀化问题和Lipschitz区域上的高阶抛物方程组的$L^p$唯一可解性。我们希望通过建立收敛速率和诸如内部以及边界的Lipschitz估计、H\"older估计和$W^{1,p}$ 估计这样的一些先验估计来对均匀化问题的的边界正则性有很好的把握，特别的，我们主要研究具有快速震荡的周期或几乎周期系数的满足Robin边界条件或混合边界条件的二阶抛物方程组。同时，我们还考虑Lipschitz区域上的高阶抛物方程组的$L^p$边值问题唯一可解的$p$的最佳范围。本项目属于偏微分方程，补偿紧方法和调和分析以及泛函分析的结合。

我们建立了Lipschitz区域上的带Neumann边值的弹性力学方程组均匀化问题的$L^2$估计和$L^p$正则性问题。通过采用紧方法，对于Lipschitz区域上具有几乎周期系数的椭圆方程组的$W^{1,p}$估计，也得到了其成立的$p$的范围，对于带Neumann边值一致椭圆均匀化问题，我们证明了其在Lipschitz区域和凸区域上的$W^{1,p}$可解性，对单个方程，我们的范围是最佳的。同时我们也考虑了一簇系数同时依赖于时间和空间的二阶抛物算子的均匀化Dirichlet和Neumann边值问题，我们得到了其一致最佳收敛速率。