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Singular Solutions to the Monge-Ampere Equation

Connor R. Mooney

Title:
Singular Solutions to the Monge-Ampere Equation
Author(s):
Mooney, Connor R.
Thesis Advisor(s):
Savin, Ovidiu
Date:
Type:
Theses
Degree:
Ph.D., Columbia University
Department(s):
Mathematics
Persistent URL:
Abstract:
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.
Subject(s):
Mathematics
Item views
342
Metadata:
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Suggested Citation:
Connor R. Mooney, , Singular Solutions to the Monge-Ampere Equation, Columbia University Academic Commons, .

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