【作者】 刘海蓉；

【导师】 杨孝平；

【作者基本信息】 南京理工大学 ， 系统工程， 2013， 博士

【摘要】 次黎曼流形(Sub-Riemannian manifold),粗略地讲,就是被赋予了一个分布及此分布上的一个纤维内积的流形,当考虑的分布为整个切丛时,次黎曼流形就成为黎曼流形.次黎曼流形研究的主要来源之一是控制论；同时,次黎曼流形也被用来研究力学中的非完整系统.近些年来,众多学者对次黎曼流形作了大量研究,内容涉及分析、方程、代数、几何等领域.而Heisenberg群是一类最简单的、非平凡的次黎曼流形；同时,由于平移映射族和伸缩映射族的存在,Heisenberg群具有丰富的内蕴结构,而且其几何结构与欧氏空间有本质的区别,因此开展Heisenberg群上的几何和分析问题的研究具有很重要的理论意义和实际价值.本论文主要研究三方面内容：一是讨论Heisenberg群上H-调和函数的增长性；二是讨论Heisenberg群上一类含有奇异位势项的散度型变系数次椭圆方程弱解的唯一延拓性；三是研究Heisenberg群上次p-Laplace方程及抛物型次p-Laplace方程的粘性解的渐近平均值公式.在H-调和函数增长性方面,我们首先讨论Heisenberg群上H-调和函数的Almgren频率的性质,得到了H-调和函数的Almgren频率与其在原点消失阶的内在关系;然后证明了H-调和函数一种新的Liouville型定理：全空间上频率有界的H-调和函数必是多项式；最后应用H-调和多项式的正交性讨论其局部增长性,证明当频率为常数时,一类具有旋转不变性的H-调和函数是齐次多项式.在次椭圆方程解的唯一延拓性方面,我们讨论Heisenberg群上一类含有奇异位势项的散度型变系数次椭圆方程弱解的唯一延拓性.首先定义这类方程弱解的频率函数,证明频率函数的单调性,然后利用频率函数单调性证明双倍条件,最后证明当方程系数、位势函数满足一定条件时解具有唯一延拓性.在渐近平均值公式方面,我们首先证明Heisenberg群上函数水平最大(最小)增长方向与水平导数之间的关系,这个关系正是Heisenberg群区别于欧氏空间的集中表现；之后研究次p-Laplace方程粘性解的渐近平均值公式,证明了粘性解和渐近平均值公式的等价性,并举反例说明这类渐近平均值公式在非渐近情形下是不成立的；最后我们讨论Heisenberg群上抛物型次p-Laplace方程,证明抛物型次p-Laplace方程粘性解的一个等价定理,在其基础上用渐近平均值公式刻画抛物型次p-Laplace方程的粘性解.更多还原

【Abstract】 A sub-Riemannian manifold, roughly speaking, is a smooth manifold associated with a distribution and a fibre-inner product on it. When the distribution is the whole tangent bun-dle, then the sub-Riemannian manifold reduces to be a Riemannian manifold. Sub-Riemannian structures arise, for instance, when dealing with non-holonomic systems in mechanics and in control theory. In recent years, there are a lot of investigations on sub-Riemannian manifolds which have strong relationships with many fields such as analysis, PDE, algebra and geometry. The Heisenberg group is the most important and simplest model of sub-Riemannian manifolds. Moreover, because of the existence of the translation and dilation, the Heisenberg group has rich intrinsic structure. Therefore, the research on the Heisenberg group has great significance in theory and application. This thesis deals with problems in three aspects:one is to discuss the growth of H-harmonic functions on the Heisenberg group; the second, we study unique contin-uation property of solutions of sub-elliptic equations with singular potential on the Heisenberg group; finally, we discuss asymptotic mean value formulae for the viscosity solutions to sub-p-Laplace equations and to sub-p-Laplace parabolic equations on the Heisenberg group.In the first aspect, the properties of Almgren’s frequency for H-harmonic functions are discussed. The relationship between frequency and the vanishing order at the origin is obtained. A new Liouville type Theorem is proved, that is, an H-harmonic function on the Heisenberg group with bounded frequency is a polynomial. Finally, we show that a class of H-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.In the second aspect, the Almgren’s frequency for solutions of a class of sub-elliptic equa-tions with singular potential on the Heisenberg group is introduced. The monotonicity property of the frequency is established and a doubling condition is obtained. Consequently, a quantita-tive proof of the unique continuation property for these equations is given.In the third aspect, we first characterize the directions of horizontal maximum (minima) of a function in terms of the horizontal gradient. Secondly, we characterize sub-p-harmonic functions on the Heisenberg group in terms of an asymptotic mean value property; Moreover, we construct an example to show that these formulae do not hold in non-asymptotic sense. Fi-nally, we derive two equivalent definitions of the viscosity solutions to sub-p-Laplace parabolic equations on the Heisenberg group, and characterize the viscosity solutions of sub-p-Laplace parabolic equations in terms of an asymptotic mean value formula.更多还原