Research On Eigenvalues For Finitely Degenerate Elliptic Operators

In this thesis,we primarily focus on the eigenvalue problems for a class of selfadjoint degenerate elliptic operators ?_X=-?_j=1^mX*_jX_j generated by the system of real smooth vector fields X=(X1,X2,· · ·,Xm)which satisfying the H?rmander condition with H?rmander index Q ? 2.To illustrate,there are estimating Dirichlet eigenvalues for a class of generalized Grushin type degenerate elliptic operators,investigating the properties of Dirichlet heat kernel and the estimation of Dirichlet eigenvalues for general finitely degenerate operator on a bounded domain with H?rmander condition.After that,we study the properties of heat kernel and the estimate of eigenvalues for general finitely degenerate operator with H?rmander condition on compact manifolds.Finally,we evaluate the bounds of Dirichlet eigenvalues for a class of higher order degenerate elliptic operators.The main contents consist of the following five chapters:In Chapter 1,we present the basic theories concerning finitely degenerate elliptic operators,and then a review of the historical and latest results on eigenvalue problems for classical elliptic operators and degenerate elliptic operators.Finally,we state our main results and give some preliminaries.In Chapter 2,we estimate the lower bound of Dirichlet eigenvalue for a class of generalized Grushin type degenerate elliptic operators.From Fourier transform and subelliptic estimates and combining with suitable projection skills,we obtain the lower bound of Dirichlet eigenvalues for the generalized Grushin type degenerate elliptic operators,in which the k-th Dirichlet eigenvalue possesses the lower bound of polynomial increasing in k with the order relating to the generalized Métivier index of vector fields within the domain.In Chapter 3,we devote to study the Dirichlet eigenvalue problems for the general self-adjoint finitely degenerate elliptic operators on a bounded domain with H?rmander condition.By the weighted Sobolev inequality,we prove the existence and uniform upper bound of Dirichlet heat kernel for finitely degenerate operators.Then,the diagonal asymptotic estimate for degenerate Dirichlet heat kernel is obtained via the maximum principle of degenerate heat equations.From uniform upper bound and Tauberian formula,we obtain the lower bound estimate and an asymptotic formula of Dirichlet eigenvalues for finitely degenerate elliptic operators.Based upon these estimations,we prove the lower bound estimate of the k-th Dirichlet eigenvalue which is polynomial increasing in k with the order relating to the generalized Métivier index of vector fields within the domain.With the asymptotic formula,we claim that this order which depends on the generalized Métivier index is optimal in the case of a certain weak hypothesis.Moreover,with respect to these general degenerate elliptic operators,we get the estimation of upper bound of Dirichlet eigenvalues which would be optimal in some cases in which the lower bounds of eigenvalues will have the same increasing order with the upper bounds in k.The Chapter 4 is still related to the eigenvalue problems for the general self-adjoint finitely degenerate elliptic operators with H?rmander condition,but it is defined on the compact manifold without boundary.Using the heat semigroup operators,subelliptic estimates and spectral theory,we prove the existence and relevant properties of degenerate heat kernel of finitely degenerate elliptic operators on the compact manifold without boundary.In addition,it follows from the results of Gaussian bound for degenerate heat kernel and the estimation of volume of Carnot-Carathéodory balls that there exists the uniform upper bound of degenerate heat kernel.On account of this uniform upper bound and some asymptotic formulas,also,we get the lower bound estimates and asymptotic formulas of eigenvalues for finitely degenerate elliptic operators,which is consistent with the one equipped with Dirichlet boundary.We end up with the investigation on Dirichlet eigenvalue problems for higher order degenerate elliptic operators.Starting from the maximal hypoellipticity of these degenerate operators,the corresponding subelliptic estimates are proved.Then the Fourier transform leads to the discovery of the lower bounds of the sum of first k eigenvalues.