Regularities for solutions to the Lₚ dual Minkowski problem for unbounded closed sets

Recently, the dual Minkowski problem for unbounded closed convex sets in a pointed closed convex cone was proposed and a weak solution to the corresponding Monge-Amp\`ere equation is provided. In this paper, we consider the regularities of solutions to this problem. More generally, we consider the Lₚ dual Minkowski problem for unbounded closed convex sets which amounts to solving a Dirichlet problem for a class of Monge-Amp\`ere type equations. Our main purpose is to show the existence, regularity and uniqueness of solution to this problem in case p 1 by studying variational properties for a family of Monge-Amp\`ere functinals. We also discuss the existence and optimal global H\"older regularity in case p<1 and q n.