Regularities for solutions to the $L_p$ dual Minkowski problem for unbounded closed sets

Abstract: Recently, the $L_p$ dual Minkowski problem for unbounded closed convex sets in a pointed closed convex cone was proposed and a weak solution to this problem was provided. In smooth setting, this problem is equivalent to solving the Dirichlet problem for a class of Monge-Amp`ere type equations. In this paper, we show the existence, regularity and uniqueness of solutions to this Monge-Amp`ere type equation in the case $p\geq 1$ by studying variational properties for a family of Monge-Amp`ere functionals. Moreover, the existence and optimal global H\"older regularity in the case $p<1$ and $q\geq n$ is also be discussed.