Research On Dirichlet Eigenvalues Of Homogeneous H(?)rmander Operators

Let X1,X2,...,Xm be linearly independent smooth vector fields in Rn satisfying Hormander’s condition and a suitable homogeneity property with respect to a family of non-isotropic dilations.This kind of the vector fields are called homogeneous Hormander vector fields.Suppose that Ω is an open bounded domain in Rn containing the origin.We study the Dirichlet eigenvalue problem of homogeneous Hormander operators ΔX=∑j=1mXj2 on Ω.By utilizing subelliptic heat kernel estimates,the resolution of singularities in algebraic geometry,and employing some refined analysis involving convex geometry,we establish the explicit asymptotic behaviour λk≈k2/Q0(ln k)-2d0/Q0 as k→+∞,where λk denotes the k-th Dirichlet eigenvalue of ΔX on Ω,Q0 is a positive rational number,and d0 is a non-negative integer not lager than n-1.Furthermore,we also give the optimal bounds of index Q0,which depends on the homogeneous dimension associated with the homogeneous Hormander vector fields.This paper is organized as follows.In Chapter 1,we review the history of degenerate elliptic operators.Meanwhile,we give the definition of homogeneous Hormander vector fields and homogeneous Hormander operators.Then we introduce the historical and recent results on eigenvalue problems for classical elliptic operators and degenerate elliptic operators.Finally,we state our main results.In Chapter 2,we investigate the properties of the homogeneous Hormander vector fields.As an important geometry object,homogeneous dimension gives a classification of the homogeneous Hormander vector fields.In Chapter 3,we introduce the so-called Folland-Stein spaces,which are the natural spaces when dealing with problems related to the Hormander operators.we discuss the Poincare inequality,chain rules and dense property in this chapter.It’s worth pointing out that the dense property depends on the homogeneity of the vector fields.In Chapter 4,we construct the Dirichlet subelliptic heat kernel and the global subelliptic heat kernel of self-adjoint homogeneous Hormander operators,by means of Dirichlet forms and heat semi-groups.We first show local L∞ estimates so that we obtain the existence of heat kernels.Then,by the Gaussian bounds of the global subelliptic heat kernel,we obtain the small-time upper and lower bounds of the difference of two diagonal subelliptic heat kernels on Ω.In Chapter 5,we give the estimates for I(r):=∫Ωdx/|BdX(x,r)| when r→0+,where BdX(x,r)is the ball induced by Carnot-Caratheodory metric determined by homogeneous Hormander vector fields.We deal with this integral by the resolution of singularities in algebraic geometry and some refined analysis involving convex geometry.For specific homogeneous Hormander vector fields,we obtain a method to calculate the explicit asymptotic behaviour of I(r)exactly.In chapter 6,we give the proof of main results.By Chapter 2 and Chapter 4,we obtain the estimate of Dirichlet subelliptic heat kernel trace ∫Ω hD(x,x,t)dx,which is asymptotic to I((?))in the sense of order when t→0+.By Chapter 5,we give the explicit formula I((?))≈tQ0/2|ln t|d0(t→0+).As a result,we conclude thatλk≈k2/Q0(ln k)-2d0/Q0(t→0+).As further applications of main results,we give some classical examples in Chapter 7.Some of the examples show that the range of Q0 is the best under the assumption of homogeneous Hormander vector fields.Moreover,one of the examples indicates that Q0 may not be the integer.