Speaker: Marta Lewicka
Title: “Convex integration for the Monge-Ampere equation in two dimensions”.
Abstract:
We discuss the dichotomy of rigidity vs. flexibility for the
\begin{equation}
{\mathcal{D}et} \nabla^2 v := -\frac 12 \mbox{curl curl } (\nabla v \otimes \nabla v) = f \qquad \mbox{in } \Omega\subset\mathbb{R}^2.
\end{equation}
Firstly, we show that below the regularity threshold
This flexibility statement is a consequence of the convex integration
Secondly, we prove that the same class of very weak solutions fails the above flexibility in the regularity regime
Our interest in the regularity of Sobolev solutions to the Monge-Ampere equation is motivated by the variational description of shape formation, which I will also explain in the talk.