幂零Lie群上次椭圆偏微分方程组的正则性研究

The nonlinear sub-elliptic systems considered in this project comes from sub-Riemann geometry, quantum physics and fluid mechanics, etc. Regularity of weak solutions to sub-elliptic systems is one of the focus in the study of PDE. This project is concerned with optimal partial regularity of weak solutions to the sub-elliptic systems on nilpotent Lie groups, and obtain the optimal H?lder exponent and the sharp singular set..In order to overcome the difficulty due to the non-commutativity of vector fields, the idea here is as follows: To connect the regularity of weak solutions to sub-elliptic systems with the harmonic approximation method on the nilpotent Lie groups. That is to say, one obtains the optimal H?lder continuity of the weak solutions by establishing and applying the harmonic approximation theory on the nilpotent Lie groups..With regard to the harmonic approximation theory, we will study several A-harmonic approximation lemmas under super-quadratic growth conditions and sub-quadratic growth conditions, respectively, which generalizes the harmonic approximation theory on the Euclidean space to the case of non-commutative nilpotent Lie groups..As for the optimal partial regularity, by using harmonic approximation methods instead of the classical direct method, we will study the sub-elliptic systems with Dini continuous coefficients and with VMO discontinuous coefficients, respectively, and establish the optimal H?lder continuity of the weak solutions. It will help to make a connection between the partial regularity of weak solutions to the sub-elliptic systems and the harmonic approximation theory on the nilpotent Lie groups.

本项目研究的非线性次椭圆方程组来源于次Riemann几何、量子物理以及流体力学等领域。次椭圆方程组弱解的正则性是偏微分方程研究的热点之一。本项目重点关注幂零Lie群上的次椭圆方程组，研究其弱解的最优部分正则性，得到最优H?lder指标和最佳奇异集。.为了克服向量场的非交换性带来的困难，我们采用如下思想：把次椭圆方程组弱解的正则性与幂零Lie群上的调和逼近方法联系起来，通过建立和应用幂零Lie群上的调和逼近理论，得到弱解的最优H?lder连续性。.调和逼近理论方面，分别研究超二次和次二次增长条件下的A-调和逼近引理，把欧氏空间的调和逼近理论发展到非交换幂零Lie群上。.最优部分正则性方面，利用调和逼近方法取代经典的直接法，分别研究具Dini连续和具VMO不连续系数的次椭圆方程组，建立其弱解的最优H?lder连续性，揭示幂零Lie群上的次椭圆方程组弱解的正则性与调和逼近理论的联系。

本项目研究的非线性次椭圆方程组来源于次Riemann几何、量子物理以及流体力学等领域。本项目重点研究了非交换幂零Lie群上的次椭圆方程组的弱解在非经典Hölder空间中的最优部分正则性。.. 本项目克服了幂零Lie群上的水平向量场的非交换性带来的困难，研究进展顺利。采取的主要思想是：把次椭圆方程组弱解的正则性与幂零Lie群上的调和逼近方法联系起来，通过建立和应用幂零Lie群上的调和逼近方法，得到弱解的最优Hölder连续性。目前，本项目已完成预期研究内容和目标，它包括：（1）建立幂零Lie群上的调和逼近方法；（2）应用调和逼近技巧，对幂零Lie群上具有Dini连续（弱于Hölder连续）的非线性次椭圆方程组，分别在超二次结构和次二次结构条件下，建立了其弱解的最优部分正则性；（3）对幂零Lie群上具有VMO不连续系数的非线性次椭圆方程组，建立了弱解的Hölder连续性。.. 本项目将有助于沟通和深化不同数学分支之间的联系，有助于深入探讨由一般Hörmander向量场构成的次椭圆方程组弱解的正则性。