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

Li Chen; Qiang Tu

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arXiv · 2403.00651

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

Li Chen·Qiang Tu

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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-Ampère type equations. In this paper, we show the existence, regularity and uniqueness of solutions to this Monge-Ampère type equation in the case $p\geq 1$ by studying variational properties for a family of Monge-Ampère functionals. Moreover, the existence and optimal global Hölder regularity in the case $p<1$ and $q\geq n$ is also be discussed.

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Li Chen, Qiang Tu. 2024-04-29. Regularities for solutions to the $L_p$ dual Minkowski problem for unbounded closed sets. https://arxiv.org/abs/2403.00651

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