Partial Regularity for Singular Solutions to the Monge‐Ampère Equation
By Connor Mooney. 2013
52 citations recorded by the source corpus.
We prove that solutions to the Monge‐Ampère inequality detD2u≥1 in ℝn are strictly convex away from a singular set of Hausdorff (n‐1)‐dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to det D2u = 1 with singular set of Hausdorff dimension as close as we like to n‐1. As a consequence we obtain W2,1 regularity for the Monge‐Ampère equation with bounded right‐hand side and unique continuation for the Monge‐Ampère equation with sufficiently regular right‐hand side. © 2015 Wiley Periodicals, Inc.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/c1981cf83e635af45fa3ca78aff0e84fd6f66555
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In Applied Mathematics
- Guido De Philippis
- Alessio Figalli
- Filippo Santambrogio
- Max Fathi
- Luigi Ambrosio
- Nicola Gigli
- Dario Trevisan
- Connor Mooney
In Geometry and Topology
- Guido De Philippis
- Alessio Figalli
- Filippo Santambrogio
- Connor Mooney
- Max Fathi
- Robert J. McCann
- Young-Heon Kim
- Cédric Villani
Citations in the corpus
- The Monge–Ampère equation and its link to optimal transportation (2013, background): “nvex neighborhood of the origin. Alexandrov showed in [3] that, in contrast with the above counterexamples, in two dimension every solution of detD2uis strictly convex (see also [22]). Recently, in [96], Mooney established that, in every dimension, the set of points where an Alexandrov solution of d”