数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (1): 237-247.doi: 10.1016/S0252-9602(17)30129-7

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THE HOMOGENEOUS POLYNOMIAL SOLUTIONS FOR THE GRUSHIN OPERATOR

刘海蓉   

  1. School of Science, Nanjing Forestry University, Nanjing 210037, China
  • 收稿日期:2016-10-29 修回日期:2017-11-05 出版日期:2018-02-25 发布日期:2018-02-25
  • 作者简介:Hairong LIU,E-mail:hrliu@njfu.edu.cn
  • 基金资助:

    This work is supported by National Natural Science Foundation of China (11401310), Natural Science Foundation of Jiangsu Province (BK20140965), High level talent research fund of Nanjing Forestry University (G2014022), and supported by the overseas research program of Jiangsu Province. The author is sponsored by Qing Lan Project of Jiangsu Province.

THE HOMOGENEOUS POLYNOMIAL SOLUTIONS FOR THE GRUSHIN OPERATOR

Hairong LIU   

  1. School of Science, Nanjing Forestry University, Nanjing 210037, China
  • Received:2016-10-29 Revised:2017-11-05 Online:2018-02-25 Published:2018-02-25
  • Supported by:

    This work is supported by National Natural Science Foundation of China (11401310), Natural Science Foundation of Jiangsu Province (BK20140965), High level talent research fund of Nanjing Forestry University (G2014022), and supported by the overseas research program of Jiangsu Province. The author is sponsored by Qing Lan Project of Jiangsu Province.

摘要:

In this paper, the author computes the dimension of space of homogeneous Grushin-harmonic functions, and give an orthogonal basis of them. Moreover, the author describes the nodal curves of these homogenous Grushin-harmonic basis. As an application of the orthogonal basis, the author proves a Liouville-type theorem for the Grushin operator, that is the Grushin-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.

关键词: Grushin operator, frequency function, homogeneous polynomial

Abstract:

In this paper, the author computes the dimension of space of homogeneous Grushin-harmonic functions, and give an orthogonal basis of them. Moreover, the author describes the nodal curves of these homogenous Grushin-harmonic basis. As an application of the orthogonal basis, the author proves a Liouville-type theorem for the Grushin operator, that is the Grushin-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.

Key words: Grushin operator, frequency function, homogeneous polynomial