Child Related To The Isoperimetric Inequality, Feature Set And Trace Theorems, Grushin Operator

This paper is devoted to the study of the isoperimetric inequality , characteristic sets and trace theorem related to the Grushin operator.In Chapter 1, some basic notations and results on Grushin operator are given. We also describe the problems and background studied in this paper.In Chapter 2, by virtue of variational method and Steiner symmetry, we discuss the isoperimetric inequality , give the isoperimetric set and its sharp constant in the Grushin plane;Moreover, by means of exploring the relation between Grushin balls and the solution to isoperimetric problem, we show the natural generalization of Brunn-Minkowski inequality in the Euclidean space does not hold to the geometric setting of the Grushin plane.In Chapter 3 , we discuss the characteristic sets related to Grushin operator, and give the sharp size of the characteristic sets, for the codimension 1 submanifold under the Grushin metric corresponding to a class of Grushin subelliptic operator in the Grushin plane.In Chapter 4, we make use of the method to investigate the trace inequality in the Grushin plane , for the Grushin operator dissatisfying local Hormander rank condition, and establish the trace inequality on a non characteristic surface on the boundary of a class of domains in the high-dimension Grushin vector fields.