Lower bounds of Dirichlet eigenvalues for a class of higher order degenerate elliptic operators

摘要

Let Omega be a bounded open domain in R-n with smooth boundary partial derivative Omega, X = (X-1, X-2, ... , X-m) be a system of real smooth vector fields defined on Omega and the boundary partial derivative Omega is non-characteristic for X. Denote lambda(k) as the k-th Dirichlet eigenvalue for degenerate elliptic operator L on Omega with L = (Sigma(m)(j=1) X-j(2p))(2), p = 1, then in this paper, we give a lower bound estimate of lambda(k) for the operator L by using weighted Sobolev embedding theorem and maximally hypoelliptic estimate.