# Analysis Seminar

#### Counterexamples to Sobolev Regularity for Degenerate Monge-Ampere Equations

Speaker: Connor Mooney, ETH Zurich

Location: Warren Weaver Hall 1302

Date: Thursday, May 4, 2017, 11 a.m.

Synopsis:

$$W^{2,1}$$ estimates for the Monge–Ampère equation $$\det D^2u = f$$ in $$R^n$$ were obtained by De Philippis and Figalli in the case that $$f$$ is bounded between positive constants. Motivated by applications to the semigeostrophic equation, we consider the case that $$f$$ is allowed to be zero on some set. In this case there are simple counterexamples to $$W^{2,1}$$ regularity in dimension $$n >= 3$$ that have a Lipschitz singularity.

In contrast, if $$n = 2$$ then a classical result of Alexandrov on the propagation of Lipschitz singularities implies that solutions are $$C^1$$. We will discuss a counterexample to $$W^{2,1}$$ regularity in two dimensions whose second derivatives have nontrivial Cantor part, and a related result on the propagation of Lipschitz/Log singularities which is optimal by example.