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Multiple solutions of the planar \(L_p\) dual Minkowski problem

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Abstract

We are concerned with the planar \(L_p\) dual Minkowski problem with indices p, q. Through the compactness analysis of an associated constrained variational problem in Sobolev space, the solvability of the planar \(L_p\) dual Minkowski problem and the related functional inequality are established, upon which the multiple solutions to the planar \(L_p\) dual Minkowski problem are obtained. Precisely, if \(q\ge 2\) is even, \(p<0\) and \(q-p>16\), there exist at least \([\sqrt{q-p}-2\ ]\) convex bodies whose \(L_p\) dual curvature measure is equal to the standard spherical measure in the plane, where \([\sqrt{q-p}-2\ ]\) is the integer part of \(\sqrt{q-p}-2\).

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Acknowledgements

This work was supported by NSFC under Grants nos. 12071482, 11931012, 11871386, and the Fundamental Research Funds for the Central Universities (WUT:2020IB017-011-019). The authors would like to thank the referee for helpful comments.

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Authors and Affiliations

  1. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, 430073, China

    Jiang Yongsheng

  2. Department of Mathematics, Center for Mathematical Sciences, Wuhan University of Technology, Wuhan, 430070, China

    Wang Zhengping

  3. Department of Mathematics and Statistics, Curtin University, Perth, WA, 6845, Australia

    Wu Yonghong

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  1. Jiang Yongsheng
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  2. Wang Zhengping
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  3. Wu Yonghong
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Correspondence to Wang Zhengping.

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Communicated by Andrea Malchiodi.

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Yongsheng, J., Zhengping, W. & Yonghong, W. Multiple solutions of the planar \(L_p\) dual Minkowski problem. Calc. Var. 60, 89 (2021). https://doi.org/10.1007/s00526-021-01950-6

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