Singular Solutions to the Monge-Ampere Equation
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.
Academic Commons
Mooney, Connor R.
Author
Savin, Ovidiu
Thesis advisor
Columbia University. Mathematics
Originator
Theses
English
Mathematics
text
2015
eng
2017-06-12T17:42:07Z
2017-11-10T07:37:17Z
Ph.D.
2
Mathematics
Columbia University
10.7916/D89K4955