Fields Seminar in Applied Math

Event Information Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations
14:10 on Wednesday December 02, 2015
15:00 on Wednesday December 02, 2015
Stewart Library, Fields Institute, 222 College St.
Connor Mooney
https://www.ma.utexas.edu/users/cmooney/
University of Texas at Austin

$W^{2,1}$ estimates for the Monge-Ampere equation $\det D^2u = f$ in $R^n$ were first 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 bounded but allowed to be zero on some set. In this case there are simple counterexamples to $W^{2,1}$ regularity in dimension $n\geq 3$ that have a Lipschitz singularity. In contrast, if $n = 2$ a classical theorem of Alexandrov on the propagation of Lipschitz singularities shows 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 also a related result on the propagation of Lipschitz / log(Lipschitz) singularities that is optimal by example.