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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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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DOI: https://doi.org/10.1007/s00526-021-01950-6