H型群上泰勒展开式及Hamilton-Jacobi方程的粘性解

【作者】 贾化冰；

【导师】 钮鹏程；

【作者基本信息】 西北工业大学 ， 基础数学， 2005， 硕士

【摘要】 本文研究了海森堡型群上一类Hamilton-Jacobi方程粘性解的存在性和唯一性,给出了光滑函数在海森堡型群G上的泰勒展开式和光滑函数在G×R~+上的泰勒展开式。 第一章介绍了“粘性解”的定义和发展,以及海森堡型群的有关知识。 第二章给出了光滑函数在海森堡型群G上的泰勒展开式和光滑函数在G×R~+上的泰勒展开式,也给出了光滑函数在海森堡群H_n上的泰勒展开式和光滑函数在H_n×R~+上的泰勒展开式。 第三章我们考虑G×R~+上的Hamilton-Jacobi方程 其中G表示海森堡型群,Du表示u的水平梯度。当函数H是径向的、凸的且超线性时,我们建立了该方程在连续初值条件u(p,0)=g(p)下有界粘性解的存在唯一性,其解由Hopf-Lax公式给出: 其中函数L是从H的水平Legendre变换提升到G上的,且关于Carnot-Carathéodory度量是径向的。更多还原

【Abstract】 This paper is devoted to the study of the Taylor expansions of smooth functions on groups of Heisenberg type G , the Taylor expansions of smooth functions onG× R+ , and then existence and uniqueness for viscosity solutions of a kind of partialdifferential equations on groups of Heisenberg type.In Chapter One, we introduce the definition and development of "viscosity solution" and some contents of groups of Heisenberg type.In Chapter Two, the Taylor expansions of smooth functions on groups ofHeisenberg type G and the Taylor expansions of smooth functions on G×R+ are established, while the Taylor expansions of smooth functions on the Heisenberg group Hn and the Taylor expansions of smooth functions on Hn × R+ are given.In Chapter Three, we consider Hamilton-Jacobi equationsut + H(Du) = 0in the G ×R+, where G is the groups of Heisenberg type and Du denotes the horizontal gradient of u. We establish the existence and uniqueness of bounded viscosity solutions with continuous initial data u(p,0) = g(p). When the HamiltonianH is radial, convex and superlinear, we prove that the solution is given by the following Hopf-Lax formulawhere L is the horizontal Legendre transform of H lifted to G by requiring it to be radial with respect to the Carnot-Caratheodory metric on G.更多还原