Heisenberg群上精确积分不等式及应用

It's well known that integral inequalities play the important role in the study of differential equations. This program is mainly devoted to study some classes of integral inequalities on the Heisenberg group. Main results are as follows: 1)Compute the explicit extremal functions and sharp constants of Hardy-Littlewood-Sobolev(HLS) type inequality and Moser-Trudinger type inequality on the Heisenberg group; 2)Establish a class of HLS type inequality with 0<p,q<1 and Onofri type inequality with negative exponent on the Heisenberg group, and then.discuss its extremal problems and sharp constants ; 3)On the half space of Heisenberg group, establish a class of HLS type inequality and Onofri type inequality which can be applied to study a class of sub-Laplacian equation with boudary; 4)Search the explicit extremal functions and sharp constants of Caffarelli-Kohn-Nirenberg type inequality on the Heisenberg group, which can be applied to discuss a class of PDEs with singular potential.. In the program, our main techniques include three aspects. 1) Firstly, by the group Fourier theory, an equivalence of equations will be established between the critical equation and integral equation. Secondly, combining the idea of moving plane method and moving sphere method, we will study the property such as cylindric symmetry and asymptotic behaviors of extremal function of these inequalities on the Heisenberg group. Lastly, classification results of the extremal functions are obtained. 2) Through Cayley transform, the classification problem on the Heisenberg group can be transformed as the classification problem on the CR sphere. This will help us to find the explicit extremal functions and sharp constants, and then we can obtain the result of classification.. Our program contribute to enrich the theory of PDE on the Heisenberg group and discover more geometry property of CR manifold.

众所周知，积分不等式在微分方程研究中有着重要作用。本项目拟在Heisenberg群上建立0<p,q<1时的Hardy-Littlewood-Sobolev型不等式、负指数的Onofri型不等式，在半空间上建立类似的不等式；进而讨论这些积分不等式的极值函数、最佳常数；计算Caffarelli-Kohn-Nirenberg型不等式的极值函数、最佳常数，进而讨论具奇异位势的微分方程解的存在性、特征值问题等等。拟利用群Fouier分析理论建立微分方程与积分方程的等价性，采用积分型移动平面法、移动球面法的思想讨论极值函数的对称性、渐进性，以获得极值函数的分类结果；或利用Cayley变换转换为CR球面上积分不等式问题，以寻求精确的极值函数、最佳常数。本项目有助于丰富Heisenberg群上微分方程理论，加深对CR流形几何性质的理解。

本项目主要研究了Heisenberg群上的移动球面法、紧Riemann流形上的Hardy-Littlewood-Sobolev(HLS)不等式及Yamabe问题、Caffarelli-Kohn-Nirenberg(CKN)型不等式、生化趋化系统的斑图解的定性和定量分析等问题，且取得了较好的结果。这些研究在偏微分方程、几何分析、调和分析、生物数学等学科中有着重要的研究意义。主要研究结果包括：(1)在紧Riemann流形上建立了HLS不等式（包括正指数和负指数的情形），提出一种新的紧性讨论方法，进而讨论了不等式的最佳常数和极值函数，并应用于局部平坦流形上Yamabe问题的讨论，为经典Yamabe问题给出一个新的简洁证明；(2)结合Heisenberg群与复球面间的Cayley变换，分析了Heisenberg群上的各种Kelvin变换；分析Jerison和Lee对CR Yamabe方程给出的解族，得到以任意点为心的CR反演变换，从而得到两个Li-Zhu型关键引理，为Heisenberg群上的移动球面法的应用提供了坚实的理论基础；(3)避开对称化技巧，通过直接证明Heisenberg群上的Hardy-Sobolev型不等式，从而在Heisenberg群上建立CKN型不等式；然后推广至Heisenberg型群和广义Baouendi-Grushin向量场上，获得了类似的CKN型不等式；(4)利用向量场法，讨论了与广义Greiner向量场相关的半线性方程，给出一类Liouville型定理；(5)分析了具体积填充效应和Logistic增长项的生化趋化系统，给出斑图解出现的一类充分条件；然后采用弱非线性分析，对斑图解进行了定量分析，分别在空间区域较小和较大时给出了斑图解的精确渐近表示公式，且数值仿真结果表明理论分析结果与实际符合度很高；特别，我们给出了一类具大振幅的静态斑图解，回答了马满军教授等提出的一个公开问题；(6)用度指标理论研究了具CIMA反应项的Lengyel-Epstein系统，给出斑图解存在的一类充分条件；(7)用积分形式的移动平面法研究了欧式空间上的Hénon方程、Hénon方程组以及单位球内具Hardy-Sobolev型权函数的半线性方程解的分类、对称性等问题。