The planar Lp dual Minkowski problem

Abstract

In this paper we study the Lpq-dual Minkowski problem for the case p < 0 < q. We prove for any positive smooth function f on \(\mathbb{S}^{1}\), there exists an F: ℝ+ → ℝ, such that if F(q) < p < 0 or 0 < q < −F(−p) then there is a smooth and strictly convex body solving the planar Lpq-dual Minkowski problem.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11971424 and 11571304).

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Correspondence to Weimin Sheng.

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In Memory of Professor Zhengguo Bai (1916–2015)

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Sheng, W., Xia, S. The planar Lp dual Minkowski problem. Sci. China Math. (2020). https://doi.org/10.1007/s11425-020-1682-9

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Keywords

  • Lp dual Minkowski type problem
  • elliptic PDE
  • degree theory

MSC(2010)

  • 35K96
  • 53C44