Unique continuation for differential inclusions

The Sobolev regularity of solutions to the Monge-Ampère equation in the plane can be rephrased in terms of a unique continuation property of differential inclusions. After an overview of the known results concerning the Monge-Ampère equation, I will explain this connection to differential inclusions and sketch a proof of the unique continuation property. Based on joint work with G. De Philippis and A. Guerra.