EVENT DETAILS AND ABSTRACT

Title: Counterexamples to Sobolev regularity for degenerate Monge-Ampere equations
Speaker: Connor Mooney
Speaker Info: University of Texas, Austin
Brief Description:
Special Note:
Abstract:

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