LOWER BOUNDS OF DIRICHLET EIGENVALUES FOR A CLASS OF FINITELY DEGENERATE GRUSHIN TYPE ELLIPTIC OPERATORS

doi:10.1016/S0252-9602(17)30098-X

Acta mathematica scientia,Series B ›› 2017, Vol. 37 ›› Issue (6): 1653-1664.doi: 10.1016/S0252-9602(17)30098-X

• Articles • Previous Articles Next Articles

LOWER BOUNDS OF DIRICHLET EIGENVALUES FOR A CLASS OF FINITELY DEGENERATE GRUSHIN TYPE ELLIPTIC OPERATORS

Hua CHEN¹, Hongge CHEN², Yirui DUAN², Xin HU²

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, China;
2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received:2016-08-31 Revised:2017-03-14 Online:2017-12-25 Published:2017-12-25
Supported by:
This work is partially supported by the NSFC (11631011, 11626251).

Abstract

Abstract:

Let Ω be a bounded open domain in Rⁿ with smooth boundary ∂Ω, X=(X₁, X₂, …, X_m) be a system of real smooth vector fields defined on Ω and the boundary ∂Ω is non-characteristic for X. If X satisfies the Hörmander's condition, then the vector field is finitely degenerate and the sum of square operator △X=???20170608???X_j² is a finitely degenerate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △_X on Ω.

Key words: Dirichlet eigenvalues, finitely degenerate elliptic operators, Hörmander's condition, sub-elliptic estimate, Grushin type operator

Hua CHEN, Hongge CHEN, Yirui DUAN, Xin HU. LOWER BOUNDS OF DIRICHLET EIGENVALUES FOR A CLASS OF FINITELY DEGENERATE GRUSHIN TYPE ELLIPTIC OPERATORS[J].Acta mathematica scientia,Series B, 2017, 37(6): 1653-1664.

Trendmd

References

[1] Weyl H. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math Ann, 1912, 71(4):441-479[2] Pólya G. On the eigenvalues of vibrating membranes. Proc London Math Soc, 1961, 3(1):419-433[3] Lieb E H. The Number of Bound States of One-Body Schrödinger Operators and the Weyl Problem//The Stability of Matter:From Atoms to Stars. Berlin, Heidelberg:Springer, 2001:245-256[4] Li P, Yau S T. On the Schrödinger equation and the eigenvalue problem. Commun Math Phy, 1983, 88(3):309-318[5] Cheng Q M, Yang H. Bounds on eigenvalues of Dirichlet Laplacian. Math Ann, 2007, 337(1):159-175[6] Kröger P. Estimates for sums of eigenvalues of the Laplacian. J Funct Anal, 1994, 126(1):217-227[7] Cheng Q M, Wei G. A lower bound for eigenvalues of a clamped plate problem. Cal Var Partial Differ Equ, 2011, 42(3):579-590[8] Cheng Q M, Wei G. Upper and lower bounds for eigenvalues of the clamped plate problem. J Differ Equ, 2013, 255(2):220-233[9] Melas A. A lower bound for sums of eigenvalues of the Laplacian. Proc Amer Math Soc, 2003, 131(2):631-636[10] Chen H, Luo P. Lower bounds of Dirichlet eigenvalues for some degenerate elliptic operators. Cal Var Part Differ Equ, 2015, 54(3):2831-2852[11] Chen H, Qiao R H, Luo P, et al. Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators. Sci China Math, 2014, 57(11):2235-2246[12] Chen H, Zhou Y. Lower bounds of eigenvalues for a class of bi-subelliptic operators. J Differ Equ, 2017, 262(12):5860-5879[13] Morimoto Y, Xu C J. Logarithmic Sobolev inequality and semi-linear Dirichlet problems for infinitely degenerate elliptic operators. Astérisque, 2003, 284:245-264[14] Xu C J. Regularity for quasilinear second-order subelliptic equations. Commun Pure Appl Math, 1992, 45(1):77-96[15] Métivier G. Fonction spectrale d'opérateurs non elliptiques. Séminaire Équations aux Dérivées Partielles (Polytechnique), 1976:1-14[16] Jerison D, Sánchez-Calle A. Subelliptic Second order Differential Operators//Complex analysis Ⅲ. Berlin Heidelberg:Springer, 1987:46-77

LOWER BOUNDS OF DIRICHLET EIGENVALUES FOR A CLASS OF FINITELY DEGENERATE GRUSHIN TYPE ELLIPTIC OPERATORS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 0

Metrics

Comments

Recommended 10