H型群上次Laplace方程的极大值原理和一维对称性

【作者】 王振华；

【导师】 钮鹏程；

【作者基本信息】 西北工业大学 ， 系统分析与集成， 2006， 硕士

【摘要】 本文致力于H型群上与De Giorgi猜想相联系的一维对称性的研究。在第一章,简单介绍了De Giorgi猜想的研究进展以及H型群的有关知识;在第二章,我们引入了H型群中Koranyi球上的极坐标表示,证明了H型群中次Laplace算子对径向函数的一个公式,构造并证明了算子T是一个紧算子;在第三章,我们首先证明了一个加细极大值原理,接着,利用加细极大值原理与Krein-Rutman定理证明了紧算子T具有正的特征值和特征函数;在第四章,我们结合算子T存在正的特征值与特征函数的性质,再次利用极坐标证明了一个H型群中无界域上的极大值原理;在第五章,我们利用极大值原理以及次Laplacian算子对H型群中群运算的左平移不变性,证明了H型群上与De Giorgi猜想相联系的一维对称性结果。本文将Birindelli,Prajapat在Heisenberg群上的结果推广到了更一般的H型群上,使得对De Giorgi猜想的研究进一步深化。更多还原

【Abstract】 This paper is devoted to the study of one dimensional symmetry on H-type group ,which is related to a conjecture by De Giorgi in R~n. In Chapter 1, the history of De Giorgi’ conjecture and some basic definitions on H-type group are given;In Chapter 2, the polar coordinates for the Koranyi unit sphere on H-type group are introduced . Then we prove an expression between the sub-Laplacian and the radial function, and construct a compact operator T;In Chapter 3, we provide a refined maximum principle for the sub-Laplacian L, with it and Krein-Rutman theorem to prove T has positive eigenvalue and eigenfunction;In Chapter 4, making use of the polar coordinates and the property that T has positive eigenvalue and eigenfunction, we establish a Maximum Principle on unbounded domains contained in H-type group;At last, in Chapter 5, by the fact that L is left invariant with respect to the group action 。 in H-type group and the maximum principle above, we prove one dimensional symmetry in H-type group. So, we generalize the work of Birindelli and Prajapat in Heisenberg group to H-type group, such that the study about De Giorgi’ conjecture becomes deeper.更多还原