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

摘要

In this paper, combining the p-capacity for p is an element of (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 pcapacitary 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.