Partial regularity for singular solutions to the Monge-Ampere equation

2013/04/09 by Mooney, Connor · 4 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1304.2706

Abstract

We prove that solutions to the Monge-Ampere inequality det D²u ≥ 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 W^2,1 regularity for the Monge-Ampere equation with bounded right hand side and unique continuation for the Monge-Ampere equation with sufficiently regular right hand side.

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• On the singular set of the free boundary for a Monge-Ampère obstacle problem
• Sharp Hausdorff Bounds for the Interior Singular Set of Convex k-Hessian Solutions
• Remarks on the quadratic Hessian equation

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