Marta Lewicka
University of Pittsburgh

Rigidity and flexibility for the Monge-Ampère equation

A $C^2$ solution $u$ to the Monge-Ampère equation in two dimensions with a positive Hessian determinant is necessarily convex. Similarly, if the Hessian determinant of $u$ vanishes, the graph of $u$ is necessarily developable. In these examples, convexity and developability are the two global characteristics of the given solution displaying its rigidity.

The purpose of this talk is to explore rigidity and flexibility of the weak type solutions to the Monge-Ampère equation by replacing the Hessian determinant with its other weaker variants. Following convex integration method a la Nash and Kuiper, we show that fixing the weak Hessian determinant as a regular enough distribution, its $C^1$ solutions are dense in the space of all continuous functions (and hence they do not admit the same rigidity as the stronger solutions). We will also discuss the connections with isometric immersion problem. The talk will be based on joint works with R. Pakzad (U. of Pittsburgh).
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