Heisenberg 群上非线性次椭圆抛物方程组弱解的正则性

Regularity of weak solutions is an important and hot issue in the study of partial differential equations. This research project is mainly concerned about the regularity of weak solutions to nonlinear sub-elliptic parabolic systems in Heisenberg groups: Under the assumptions of different structural conditions, optimal partial regularity will be established by applying a new idea --A-Caloric approximation method...A-Caloric Approximation Theory: We are beginning to establish A-Caloric approximation lemmas for the cases of super-quadratic growth and sub-quadratic growth in Heisenberg groups, respectively. Also we extend the classical A-Caloric approximation theory in Euclidean spaces to the non-commutative Heisenberg groups...Optimal Partial Regularity: We are going to consider nonlinear sub-elliptic parabolic systems with Hölder continuous coefficients, Dini continuous coefficients, and VMO discontinuous coefficients in the Heisenberg groups, respectively. Based on the method of A-Caloric approximation, the optimal Hölder continuity will be obtained for sub-elliptic parabolic systems with the super-quadratic case and the sub-quadratic case, respectively, including the optimal Hölder exponent and singular sets...We know that much more challenges will arise to study the regularity of weak solutions due to non-community for horizontal vector fields and the degeneration for sub-elliptic operators in the Heisenberg groups. So it is very different from the case of Euclidean spaces. This project breaks through barriers for non-community of the horizontal vector fields and nonlinearity of the systems by establishing and employing A-Caloric approximation method instead of the classical direct method, and then the optimal partial regularity is obtained in the Heisenberg groups.

弱解的正则性是偏微分方程研究的重点和热点问题。本项目重点研究海森堡群上的非线性次椭圆抛物方程组弱解的正则性：应用新思想—A-Caloric（简写AC）逼近方法，在不同结构条件下，建立弱解的最优部分正则性。..AC逼近理论方面：在海森堡群上分别建立超二次和次二次增长情形的AC逼近引理，并把欧式空间上经典的AC逼近理论发展到非交换海森堡群上。..正则性方面：分别对海森堡群上具有Hölder连续，Dini连续和VMO不连续系数的非线性次椭圆抛物方程组，应用AC逼近技巧，研究超二次和次二次增长指标下弱解的最优Hölder连续性，得到最佳Hölder指标和奇异集。..海森堡群上水平向量场的非交换性和其上次椭圆算子的退化性，加大了正则性理论研究的难度，与欧式空间情形有本质区别。本项目通过建立和应用海森堡群上的AC逼近方法取代经典直接法，克服向量场非交换性和方程的非线性带来的困难，得到最优部分正则性。

非线性次椭圆抛物方程组弱解的正则性理论是偏微分方程研究的重点问题。本项目主要研究海森堡群上的散度型非线性次椭圆抛物方程组在不同增长条件和不同系数假设条件下的弱解正则性。. 本项目通过建立和应用海森堡群上的A-Caloric逼近方法取代经典直接法，克服向量场非交换性和方程的非线性带来的困难，得到最优部分正则性结果。项目组已完成预期研究内容和目标。它包括：（1）分别建立Heisenberg群上超二次增长和次二次增长情形的A-Caloric逼近理论；（2）应用A-Caloric逼近方法，对海森堡群上具有Hölder连续系数的非线性次椭圆抛物方程组，在超二次和次二次增长指标下，建立弱解关于一阶水平梯度的最优Hölder连续性；（3）在Dini连续（弱于Hölder连续）系数假设下，建立弱解的C1部分正则性；（4）在VMO不连续系数假设下，建立弱解的Hölder连续性；（5）在二次增长条件下对由Hörmander向量场构成的散度型拟线性次椭圆方程组，建立其弱解的Hölder连续性。. 本项目研究的非线性次椭圆抛物方程组来源于次Riemann几何和量子物理等领域。本项目将有助于沟通和深化不同数学分支和其它学科之间的联系，有助于深入探讨由一般Hörmander向量场构成的次椭圆抛物方程组弱解的正则性。