PreprintarXiv licencearXiv
Global regularity for a class of degenerate/singular Monge-Ampère equations
Abstract
We study the global regularity of solutions to the homogeneous Dirichlet problem for a class of Monge-Ampère equations, which become degenerate or singular on the boundary due to the zero boundary value and the distance function. The degeneracy and singularity are described by the index - the sum of the powers of the distance function and the solution. We give a complete classification of the index, and prove that the solution is globally Hölder continuous, log-Lipschitz continuous, C1,η, C2,α and W2,p respectively, depending on the size of the index. We also obtain estimates related to the five regularity results. All these results are optimal.
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BibTeX
@misc{cui2026global,
title = {{Global regularity for a class of degenerate/singular Monge-Ampère equations}},
author = {Fan Cui and Huaiyu Jian and You Li and Yannan Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35192},
eprint = {2610.35192},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35192},
}
RIS
TY - UNPB
TI - Global regularity for a class of degenerate/singular Monge-Ampère equations
AU - Fan Cui
AU - Huaiyu Jian
AU - You Li
AU - Yannan Liu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.35192
UR - https://arxiv.org/abs/2610.35192
AB - We study the global regularity of solutions to the homogeneous Dirichlet problem for a class of Monge-Ampère equations, which become degenerate or singular on the boundary due to the zero boundary value and the distance function. The degeneracy and singularity are described by the index - the sum of the powers of the distance function and the solution. We give a complete classification of the index, and prove that the solution is globally Hölder continuous, log-Lipschitz continuous, C^1,η, C^2,α and W^2,p respectively, depending on the size of the index. We also obtain estimates related to the five regularity results. All these results are optimal.
ER -