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Smooth solutions to the \(L_p\) dual Minkowski problem

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Abstract

In this paper, we consider the \(L_p\) dual Minkowski problem by geometric variational method. Using anisotropic Gauss–Kronecker curvature flows, we establish the existence of smooth solutions of the \(L_p\) dual Minkowski problem when \(pq\ge 0\) and the given data is even. If \(f\equiv 1\), we show under some restrictions on p and q that the only even, smooth, uniformly convex solution is the unit ball.

Mathematics Subject Classification

52A38 35J20 

Notes

Acknowledgements

The authors are extremely grateful to the referee for his/her many valuable comments and suggestions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsZhejiang University of TechnologyHangzhouChina
  2. 2.Institute of MathematicsHunan UniversityChangshaChina
  3. 3.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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