非迷向Heisenberg群上凸函数的性质研究  

Properties of Convex Functions on the Anisotropic Heisenberg Group

作　　者：李虎俊[1] 王彦林[2] 徐飞[3] 李闻白[4] 

机构地区：[1]湖北职业技术学院公共课部,孝感432000 [2]西安工程大学理学院,西安710048 [3]西安工业大学计算机科学与工程学院,西安710032 [4]西北工业大学航海学院,西安710072

出　　处：《西安工业大学学报》2011年第2期184-193,共10页Journal of Xi’an Technological University

摘　　要：Hmander向量场上凸函数的性质对研究完全非线性次椭圆方程的正则性起关键作用.针对一类特殊的Hmander向量场——非迷向Heisenberg群H2(a1,a2),通过引入群上的拟距离函数,利用矩阵理论与粘性解理论证明了拟距离的凸性和无穷调和性,得到了H2(a1,a2)上Monge-Ampère测度的正定性质.证明了一个与凸函数相关的Harnack型不等式和Aleksandrov型极大值原理.研究结果可以为进一步研究一般H mander向量场上凸函数的性质和非线性次椭圆方程的正则性提供理论基础.The properties of convex functions on Homander vector fields are important in studying the regularity of fully nonlinear subelliptic equations.In this paper,aiming at a special class of Homander vector fields——anisotropic Heisenberg group H^2（a1,a2）,the convexity and infinity harmonicity of the quasidistance are proved by introducing the proper quasidistance function and using of matrices theory and viscosity solution theory.Then,the positivity of the Monge-Ampère measure on H^2（a1,a2） is obtained.Finally,as an interesting application of these results,a Harnack-type inequality and a maximum principle of Aleksandrov-type for convex functions are proved.The results are expected to provide some theoretical basis for further study of the properties of convex functions and the regularity of nonlinear subelliptic equations on general Homander vector fields.

关 键 词：非迷向Heisenberg群 凸函数 拟距离 Monge-Ampère测度 Harnack型不等式 Aleksandrov型极大值原理 

分 类 号：O174.13[理学—数学]