2015 Theses Doctoral

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 𝐷²𝑢 = 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.