向量场构成的退化偏微分方程的Hölder正则性

Degenerate equations of vector fields appeare in sub-Riemann geometry, harmonic analysis, geometry analysis, quantum physics, financial mathematics, random processes. The study of regularity to degenerate equations is very interesting and important..The project is to establish Hölder regularity of weak solutions to degenerate equations with bounded coefficients constructed by Hörmander’s vector fields involving drift (these equations contain degenerate elliptic equations and parabolic equations on homogeneous groups, degenerate elliptic equations and parabolic equations of Hörmander’s vector fields without drift), to obstacle problems of degenerate equations and degenerate equations with singular potential..The main difficulties in the project are that one can not derive the corresponding Sobolev inequalities with respect to Hörmander’s vector fields with drift and the metric measure space induced by Hörmander’s vector fields with drift is not a homogeneous space. To overcome these difficulties, we prove Sobolev inequalities and a Fefferman-Phong inequality for weak solutions and provide some analysis properties in the local homogeneous space caused by Hörmander’s vector fields with drift. .Our research will enrich regularity theory and methods in studying partial differential equations of vector fields, reveal essential connotation to these equations and lay a solid foundation for dealing with nonlinear partial differential equations of vector fields. The method by using the Sobolev and Fefferman-Phong inequalities of weak solutions to follow regularity to degenerate equations will be a new approach for well posedness to nonlinear partial differential equations of vector fields.

向量场构成的退化方程大量出现在次黎曼几何,调和分析,几何分析,量子物理,金融数学,随机过程等领域,对其正则性的研究具有重要的理论意义..本项目针对由Hörmander向量场(含漂移向量场)构成的,具有界可测系数的退化方程, 退化方程障碍问题和带奇异位势的退化方程建立弱解的Hölder正则性..本项目遇到的主要困难是不能对Hörmander向量场(含漂移向量场)建立Sobolev不等式,且这些向量场诱导的度量测度空间不再是齐型空间,而是一类非齐型空间-局部齐型空间.为此我们证明与退化方程弱解相关的Sobolev不等式和Fefferman-Phong不等式;研究局部齐型空间的分析性质,发展De Giorgi和Moser迭代获得正则性..本项目的研究将丰富向量场型偏微分方程的正则性理论及证明方法,为研究非线性向量场型偏微分方程奠定必要的基础。

本项目的研究背景：.本项目研究的由向量场构成的退化方程大量出现在次黎曼几何, 调和分析, 几何分析, 量子物理等领域, 对其正则性的研究有重要的理论意义。.本项目主要研究内容：.主要研究内容包括研究了Heisenberg群上带漂移项的拟线性退化椭圆方程的正则性；建立了与向量场相关的带权对数型Hardy-Sobolev不等式; 证明了在-增长条件下一类非一致非线性退化椭圆方程组弱解的弱可微性; 给出了带漂移项的非线性不连续次椭圆方程组弱解的部分Hölder正则性和弱解水平梯度的部分Morrey正则性；对一般增长条件下的非线性不可微退化椭圆方程组建立了弱解梯度的分数阶估计。.本项目取得的重要研究成果：.取得的重要研究成果包括.1 Heisenberg群上带漂移项的拟线性退化椭圆方程的正则性.2 与向量场相关的带权对数型Hardy-Sobolev不等式; .3 在-增长条件下非一致非线性退化椭圆方程组弱解的弱可微性; .4 带漂移项的非线性不连续次椭圆方程组弱解的部分Hölder正则性和弱解水平梯度的部分Morrey正则性。.5具一般增长条件的非线性不可微退化椭圆方程组弱解梯度的分数阶估计.本项目的研究意义：.本项目的研究丰富了向量场型偏微分方程的正则性理论, 揭示了这些方程的内在本质性质, 为研究非线性向量场型偏微分方程奠定了基础。