Hörmander向量场构成的非线性退化次椭圆方程组弱解的正则性

退化椭圆方程的正则性是偏微分方程理论研究的核心。本项目重点研究由Hörmander向量场构成的二阶非线性退化次椭圆方程组：发展并应用调和逼近思想，在不同增长情形下，建立弱解的正则性，包括最佳奇异集和最优Hölder指标。..调和逼近理论：利用向量场上的函数表示公式、函数空间嵌入性质和庞加莱不等式等主要工具，在Carnot-Carathéodory（简称C-C）空间上分别建立L^p（p>2）形式和带辅助函数L^2形式的调和逼近引理。..部分正则性：利用向量场的提升逼近定理和C-C空间上的调和逼近思想，克服向量场非交换性和非齐次性，以及方程组退化性带来的困难，分别研究具有Hölder、Dini和VMO系数的次椭圆方程组，在超二次和次二次结构条件下，建立弱解的Hölder连续性和Morrey正则性。. ..本项目有助于丰富次椭圆方程组的正则性理论，以及促进次黎曼几何和非交换群上调和分析的发展。

Regularity for degenerate elliptic equations is the core of the theory of partial differential equations. This project focuses on the second order nonlinear degenerate sub-elliptic systems of PDEs composed of Hörmander’s vector fields: Based on the idea of harmonic approximation, some regularity results for weak solutions are established under different exponents of growth condition, involving the optimal singular set and the sharp Hölder index...Harmonic Approximation Theory: By employing the representation formula for certain functions, embedding property of spaces and Poincaré type inequality etc, associated with Hörmander’s vector fields, we shall begin to establish the harmonic approximation lemma in L^p(p>2) form, and L^2 form with the auxiliary function in Carnot-Carathéodory (abbreviation C-C) space, respectively...Partial Regularity: We are going to study degenerate sub-elliptic systems with Hölder, Dini and VMO coefficients related to Hörmander’s vector fields, respectively. On the basis of the idea of harmonic approximation in the C-C space and the lifting approximation theorem for the vector fields, we overcome challenges due to the non-commutativity and non-homogeneity for Hörmander’s vector fields, and the degeneracy of sub-elliptic systems. Then Hölder continuity and Morrey regularity are established under the assumptions of super-quadratic structure conditions, and sub-quadratic structure cases, respectively...This project contributes to enriching the regularity theory of sub-elliptic systems structured on Hörmander’s vector fields, and is helpful to promote the development of sub-Riemannian geometry and harmonic analysis on non-Able groups.