Abstract
We construct convex functions on \(\mathbb {R}^3\) and \(\mathbb {R}^4\) that are smooth solutions to the Monge–Ampère equation
away from compact one-dimensional singular sets, which can be Y-shaped or form the edges of a convex polytope. The examples solve the equation in the Alexandrov sense away from finitely many points. Our approach is based on solving an obstacle problem where the graph of the obstacle is a convex polytope
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Acknowledgements
The author is grateful to Tianling Jin and Jingang Xiong for asking the question that motivated this research and to Richard Schoen for a helpful discussion. The research was supported by NSF Grant DMS-1854788.
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Mooney, C. Solutions to the Monge–Ampère Equation with Polyhedral and Y-Shaped Singularities. J Geom Anal 31, 9509–9526 (2021). https://doi.org/10.1007/s12220-021-00615-2
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DOI: https://doi.org/10.1007/s12220-021-00615-2