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Critical points and level sets of Grushin-Harmonic functions in the plane

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Abstract

This paper concerns the critical points and the level sets of solutions of the Grushin equation in the plane. After exactly establishing descriptions about the critical points of the homogeneous Gruhin-harmonic polynomials and investigating the local geometric properties of the level sets near these critical points, we prove that the critical points of solutions of the Grushin equation are isolated and each critical point has finite multiplicity. We further estimate the numbers of interior critical points of solutions of the Dirichlet boundary value problem.

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Acknowledgments

This work was done while the first author was visiting at The Institute of Mathematical Sciences of The Chinese University of Hong Kong. She would like to thank the institution and is very grateful to Professor Zhouping Xin for his invitation. The research of the first author was supported by the National Natural Science Foundation of China (No. 12071219, No. 12026432).

The research of the second author was supported by the National Natural Science Foundation of China (No. 11971229).

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Authors and Affiliations

  1. School of Science, Nanjing Forestry University, Nanjing, 210037, China

    Hairong Liu

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Xiaoping Yang

Authors
  1. Hairong Liu
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  2. Xiaoping Yang
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Correspondence to Hairong Liu.

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Liu, H., Yang, X. Critical points and level sets of Grushin-Harmonic functions in the plane. JAMA 143, 435–460 (2021). https://doi.org/10.1007/s11854-021-0151-x

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  • DOI: https://doi.org/10.1007/s11854-021-0151-x

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