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Singular Solutions to the Monge-Ampere Equation

Mooney, Connor R.

This thesis contains the author's results on singular solutions to the Monge-Ampere equation \det D^2u = 1. We first prove that solutions are smooth away from a small closed singular set of Hausdorff (n-1)-dimensional measure zero. We also construct solutions with a singular set of Hausdorff dimension n-1, showing that this result is optimal. As a consequence we obtain unique continuation for the Monge-Ampere equation. Finally, we prove an interior W^{2,1} estimate for singular solutions, and we construct an example to show that this estimate is optimal.

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More About This Work

Academic Units
Mathematics
Thesis Advisors
Savin, Ovidiu
Degree
Ph.D., Columbia University
Published Here
April 10, 2015