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The Lp Minkowski problem for q-capacity

Part of: General convexity Higher-dimensional theory

Published online by Cambridge University Press:  21 September 2020

Zhengmao Chen
Zhengmao Chen*
Affiliation:
LCSM (Ministry of Education), College of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, People's Republic of China (zhengmaochen@aliyun.com)
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Abstract

In the present paper, we first introduce the concepts of the Lpq-capacity measure and Lp mixed q-capacity and then prove some geometric properties of Lpq-capacity measure and a Lp Minkowski inequality for the q-capacity for any fixed p ⩾ 1 and q > n. As an application of the Lp Minkowski inequality mentioned above, we establish a Hadamard variational formula for the q-capacity under p-sum for any fixed p ⩾ 1 and q > n, which extends results of Akman et al. (Adv. Calc. Var. (in press)). With the Hadamard variational formula, variational method and Lp Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the q-capacity which extends some beautiful results of Jerison (1996, Acta Math.176, 1–47), Colesanti et al. (2015, Adv. Math.285, 1511–588), Akman et al. (Mem. Amer. Math. Soc. (in press)) and Akman et al. (Adv. Calc. Var. (in press)). It is worth mentioning that our proof of Hadamard variational formula is based on Lp Minkowski inequality rather than the direct argument which was adopted by Akman (Adv. Calc. Var. (in press)). Moreover, as a consequence of Lp Minkowski inequality for q-capacity, we get an interesting isoperimetric inequality for q-capacity.

Keywords

q-CapacityLpq-capacity measureLp Minkowski inequalityLp Minkowski problemVariational methodHadamard variational formula

MSC classification

Primary: 31B15: Potentials and capacities, extremal length
Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52A39: Mixed volumes and related topics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 4 , August 2021 , pp. 1247 - 1277
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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