Lower bounds of eigenvalues for a class of bi-subelliptic operators

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:262

Article

Chen, Hua^1,2  Zhou, Yifu^1,2 

[1] Wuhan Univ, Sch Math & Stat, Wuhan 430072, Peoples R China

[2] Wuhan Univ, Computat Sci Hubei Key Lab, Wuhan 430072, Peoples R China

关键词: Eigenvalues;    Finitely degenerate elliptic operators;    Hormander's condition;    Metivier's condition;    Subelliptic estimate;    Bi-subelliptic operator;   

DOI  :  10.1016/j.jde.2017.02.018

来源: Elsevier

【 摘 要 】

Let Omega be a bounded open domain in R-n with smooth boundary and X = (X-1, X-2, . . ., X-m) be a system of real smooth vector fields defined on Omega with the boundary partial derivative Omega which is non-characteristic for X. If X satisfies the Hormander's condition, then the vector fields are finitely degenerate and the sum of square operators Delta(X) = Sigma(m)(i=1) X-i(2) is a subelliptic operator. Let lambda(k) be the k-th eigenvalue for the bi-subelliptic operator Delta(2)(X) on Omega. In this paper, we introduce the generalized Maivier's condition and study the lower bounds of Dirichlet eigenvalues for the operator Delta(2)(X) on some finitely degenerate systems of vector fields X which satisfy the Hormander's condition or the generalized Metivier's condition. By using the subelliptic estimates, we shall give a explicit lower bound estimates of lambda(k) which is polynomial increasing ink with the order relating to the Hdrmander index or the generalized Metivier index. (C) 2017 Elsevier Inc. All rights reserved.

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