H Group Last Laplace Equation Maximum Principle And The One-dimensional Symmetry

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.