The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems

Han Hong Memorial University of Newfoundland Deping Ye Memorial University of Newfoundland Ning Zhang University of Alberta

Analysis of PDEs Functional Analysis Convex and Discrete Geometry mathscidoc:1805.03001

Calculus of Variations and Partial Differential Equations, 57, 5, 2018.2
In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_{\phi}$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized $p$-capacitary $L_q$ Minkowski problems with $q>1$ for both discrete and general measures.
Brunn-Minkowski inequality, M-addition, Minkowski inequality, Minkowski problem, mixed p- capacity, Orlicz addition of convex bodies, Orlicz-Brunn-Minkowski theory, Orlicz Minkowski problem, p-capacity
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@inproceedings{han2018the,
  title={The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems},
  author={Han Hong, Deping Ye, and Ning Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180529232317063124090},
  booktitle={Calculus of Variations and Partial Differential Equations},
  volume={57},
  pages={5},
  year={2018},
}
Han Hong, Deping Ye, and Ning Zhang. The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems. 2018. Vol. 57. In Calculus of Variations and Partial Differential Equations. pp.5. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180529232317063124090.
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