Taylor Expansions And The Hamilton-jacobi Equations, Viscosity Solutions Of The H Group

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.