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

2017/03/04 by Han Hong, Hong, Han, Deping Ye +3 · 1 citation
Mathematics · #31B15 #35J60 #52A20 #52A39 #52B45 #53A15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #math.AP #math.FA #math.MG #msc:31B15 #msc:35J60 #msc:52A20 #msc:52A39 #msc:52B45 #msc:53A15

paper · pdf · doi:10.48550/arxiv.1703.01458

Some modifications were made to fix the gap

arxiv created 2017/11/19 · arxiv updated 2017/11/21

Abstract

In this paper, combining the p-capacity for p∈ (1, n) with the Orlicz addition of convex domains, we develop the p-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz L_ϕ 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.

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