Folland-Stein算子和Kohn Laplace算子的二次多项式算子的低阶特征值不等式

数学物理学报 ›› 2014, Vol. 34 ›› Issue (5): 1133-1143.

Folland-Stein算子和Kohn Laplace算子的二次多项式算子的低阶特征值不等式

孙和军

南京理工大学理学院南京 210094

收稿日期:2013-06-20 修回日期:2014-03-10 出版日期:2014-10-25 发布日期:2014-10-25
基金资助:
国家自然科学基金(11001130)资助.

Inequalities for Lower Order Eigenvalues of the Folland-Stein Operator and Quadratic Polynomial Operator of the Kohn Laplacian

SUN He-Jun

Department of Applied Mathematics, College of Science, Nanjing University of Science and Technology, Nanjing 210094

Received:2013-06-20 Revised:2014-03-10 Online:2014-10-25 Published:2014-10-25
Supported by:
国家自然科学基金(11001130)资助.

摘要/Abstract

摘要：

研究了Heisenberg群上的Folland-Stein算子和Kohn Laplace算子的二次多项式算子的Dirichlet特征值问题, 建立了低阶特征值的一些不等式.

关键词: 特征值, Heisenberg群, Folland-Stein算子, Kohn Laplace算子

Abstract:

In this paper, we investigate the Dirichlet eigenvalue problems of the Folland-Stein operator and quadratic polynomial operator of the Kohn Laplacian on the Heisenberg group. We establish some inequalities for their lower order eigenvalues.

Key words: Eigenvalue, Heisenberg group, Folland-Stein operator, Kohn Laplacian

中图分类号:

35P15

孙和军. Folland-Stein算子和Kohn Laplace算子的二次多项式算子的低阶特征值不等式[J]. 数学物理学报, 2014, 34(5): 1133-1143.

SUN He-Jun. Inequalities for Lower Order Eigenvalues of the Folland-Stein Operator and Quadratic Polynomial Operator of the Kohn Laplacian[J]. Acta mathematica scientia,Series A, 2014, 34(5): 1133-1143.

参考文献

[1] Kohn J J. Boundaries of Complex Manifolds//Aeppli A. Proc Conf Complex Analysis (Minneapolis, 1964). Berlin: Springer Verlag, 1965: 81--94

[2] Jerison D S. The Dirichlet problem for the Kohn Laplacian on the Heisenberg group I. J Funct Anal, 1981, 43: 97--142

[3] Folland G B, Stein E M. Estimates for the ∂_b complex and analysis on the Heisenberg group. Comm Pure Appl Math, 1974, 27: 429--522
[4] Stein E M. Harmonic Analysis. Princeton: Princeton Univ Press, 1993

[5] Luo X B, Niu P C. On the Dirichlet eigenvalue problem of the Folland-Stein operator on the Heisenberg group. Acta Math Sci, 1997, 17(S1): 70--75

[6] Luo X B, Niu P C. The spectrum of the Kohn Laplace operator. Chin Ann Math, 1999, 20A: 229--234

[7] Niu P C, Zhang H Q. Payne-P\'{o}lya-Weinberger type inequalities for eigenvalues of nonelliptic operators. Pacific J Math, 2003, 208: 325--345

[8] Soufi A El, Harrell II E M, Ilias S. Universal inequalities for the eigenvalues of Laplace and Schr\"{o}dinger operators on submanifolds. Trans Amer Math Soc, 2009, 361: 2337--2350

[9] Friedrichs K O. Spectral Theory of Operators in Hilbert Space. New York: Springer Verlag, 1980

[10] Guo D H. Eigenvalue problem for a class of left invariant operators on Heisenberg group. J Beijing Norm Univ, 1992, 28: 287--292

[11] Levine H A, Protter M H. Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity. Math Meth Appl Sci, 1985, 7: 210--222

[12] Huang G Y, Chen W Y. Inequalities of eigenvalues for bi-Kohn Laplacian on Heisenberg group. Acta Math Sci, 2010, 30(1): 125--131

[13] Ilias S, Makhoul O. Universal inequalities for the eigenvalues of a power of the Laplace operator. Manuscripta Math, 2010, 132: 75--102

[14] Sun H J, Qi X R. Universal inequalities for eigenvalues of quadratic polynomial operator of the Kohn Laplacian. Math Inequal Appl, 2012, 12(3): 559--569

[15] Payne L E, P\'{o}lya G, Weinberger H F. On the ratio of consecutive eigenvalues. J Math Phys, 1956, 35: 289--298

[16] Ashbaugh M S, Benguria R D. More bounds on eigenvalue ratios for Dirichlet Laplacians in n dimension. SIAM J Math Anal, 1993, 24: 1622--1651

[17] Ashbaugh M S. Isoperimetric and Universal Inequalities for Eigenvalues//Davies E B, Safarov Yu. Spectral Theory and Geometry, London Math Soc, Lecture Notes, Vol 273. Cambridge: Cambridge Univ Press, 1999: 95--139

[18] Cheng Q M, Ichikawa T, Mametsuka S. Inequalities for eigenvalues of Laplacian with any order. Comm Contemp Math, 2009, 11(4): 639--655

[19] Cheng Q M, Huang G Y, Wei G X. Estimates for lower order eigenvalues of a clamped plate problem. Calc Var PDE, 2010, 38: 409-416

[20] Sun H J, Cheng Q M, Yang H C. Lower order eigenvalues of Dirichlet Laplacian. Manuscripta Math, 2008, 125: 139--156

[21] Sun H J, Zeng L Z. Universal Inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian. Acta Math Sinica, 2013, 29(11): 2209--2218