齐次群上退化椭圆偏微分方程解的Schuader估计

The regularity for the weak solutions is always one of the hotspots in partial differential equations. This project will mainly study the Schauder estimates for the solutions of degenerate elliptic equations of second order. The following question will be considered: when the non-homogeneous term is Dini continuous, how to get the estimates of the Hölder norms for the solutions of the degenerate elliptic equations on homogeneous group. However, the progress on this kind of problems is little because of the non-commutative properties of the vector fields on homogeneous group and the degeneration of elliptic equations. It’s difficult to apply the classical methods for getting Schauder estimates to this problem. This project will develop the classical perturbation methods by using the technical in partial differential equations to study the Schauder estimates of the solutions to degenerate elliptic equations under the assumption that the non-homogeneous term is a Dini continuous function. Moreover, some new methods for the research of this kind of problems will be provided.

偏微分方程解的正则性一直是偏微分方程理论研究中的热点问题之一。本项目将重点研究齐次群上退化二阶椭圆方程解的Schauder估计。拟解决的问题是：当非齐次项为Dini连续函数时，探索研究齐次群上退化二阶椭圆偏微分方程解的Hölder模估计。齐次群上向量场的不可交换性及椭圆方程的退化性使得这类方程解的正则性理论研究进展较少，经典的Schauder估计方法也很难直接应用于这类问题。本项目拟结合偏微分方程的技巧发展现有的摄动方法来研究具有Dini连续非齐次项的退化二阶椭圆偏微分方程解的Schauder估计，同时为研究此类问题提供一些新方法。

我们研究了齐次群上退化二阶椭圆偏微分方程解的Schauder 估计,从而获得了退化椭圆方程解的 Hölder 模与非齐次项函数Dini 模之间的关系.主要是结合偏微分方程的技巧，利用平均值定理，Taylor 公式及调和函数导数的先验估计等研究了典型的齐次群-- Heisenberg 群中 Kohn-Laplace 算子的 Schauder 估计。另外，我们还利用奇异积分理论以及适当的不等式，研究了Heisenberg 群中具有 VMO 系数的退化椭圆方程解的 Morrey 正则性; 并在可控增长条件和自然增长条件下，研究了退化抛物方程组弱解的 L^p 估计。总之，本项目帮助我们基本完成了研究计划，还带动了其他研究课题。