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Mathematics > Analysis of PDEs
Title: Regularities for solutions to the $L_p$ dual Minkowski problem for unbounded closed sets
(Submitted on 1 Mar 2024 (v1), last revised 29 Apr 2024 (this version, v2))
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.
Submission history
From: Qiang Tu [view email][v1] Fri, 1 Mar 2024 16:33:00 GMT (28kb)
[v2] Mon, 29 Apr 2024 08:12:10 GMT (24kb)
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