# MathSciDoc: An Archive for Mathematician ∫

#### Analysis of PDEsFunctional AnalysisConvex 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
@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.