Some Properties Of Solutions To Several Partial Differential Equations On The Heisenberg Group

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.