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Critical points of solutions for the mean curvature equation in strictly convex and nonconvex domains

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Abstract

In this paper, we mainly investigate the set of critical points associated to solutions of the mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce that the mean curvature equation has exactly one nondegenerate critical point in a smooth, bounded and strictly convex domain of ℝn(n ≥ 2). Secondly, we study the geometric structure about the critical set K of solutions u for the constant mean curvature equation in a concentric (respectively an eccentric) spherical annulus domain of ℝn(n ≥ 3), and deduce that K consists (respectively does not consist) of a rotationally symmetric critical closed surface S. In fact, in an eccentric spherical annulus domain, K is made up of finitely many isolated critical points (p1, p2, …, pl) on an axis and finitely many rotationally symmetric critical Jordan curves (C1, C2,…,Ck) with respect to an axis.

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Acknowledgement

The first author is very grateful to his advisor, Professor Xiaoping Yang, for his expert guidance and useful conversations.

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Authors and Affiliations

  1. School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China

    Haiyun Deng

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

    Hairong Liu & Long Tian

Authors
  1. Haiyun Deng
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  2. Hairong Liu
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  3. Long Tian
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Corresponding author

Correspondence to Haiyun Deng.

Additional information

The work is supported by the National Natural Science Foundation of China (No. 11401307, No. 11401310), the High Level Talent Research Fund of Nanjing Forestry University (G2014022) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17 0321)

The first author is fully supported by China Scholarship Council(CSC) for visiting Rutgers University (201806840122).

The second author is sponsored by Qing Lan Project of Jiangsu Province.

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Deng, H., Liu, H. & Tian, L. Critical points of solutions for the mean curvature equation in strictly convex and nonconvex domains. Isr. J. Math. 233, 311–333 (2019). https://doi.org/10.1007/s11856-019-1906-2

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