Singular Solutions to the Monge-Ampere Equation.

摘要

This thesis contains the author's results on singular solutions to the Monge-Ampere equation det D2 u = 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 W2,1 estimate for singular solutions, and we construct an example to show that this estimate is optimal.