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.


\(W^{2,1}\) 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.