This project focuses on qualitative properties of solutions to certain degenerate elliptic partial differential equations arising in different areas of mathematics. Some of the problems under investigation include fourth-order Monge-Ampere-type equations, nonlocal free boundary and phase transition problems, and degenerate fully nonlinear equations of concave type. One recurrent question is the smoothness of the solutions or their level sets and their behavior near singular points. For example, the project will investigate the smoothness of Lipschitz thin free boundaries and the dimensions of their singular sets, the boundary behavior of solutions of Monge-Ampere equations with degenerate boundary conditions, and the rigidity of level sets in nonlocal phase transitions.
Partial differential equations are used to describe a large variety of physical phenomena. At the same time, they contribute to the development of different branches of mathematics such as differential geometry, topology, probability, and algebraic geometry. In particular, Monge-Ampere equations occur naturally in the mathematical formulation of optimal transportation problems, many of them within the realm of our daily lives: traffic network planning in cities, internet traffic optimization, blood vessel branching in the human venous system, and meteorological fluid dynamics. Such equations also frequently turn up in other areas of mathematics, like Riemannian or conformal geometry. Nonlocal free boundary and phase transition problems appear in several areas of science, including fluid mechanics, plasma physics, and semiconductor theory. Thin free boundaries are relevant to our understanding of flame propagation and also in the propagation of surfaces of discontinuities. The final purpose of this project is to gain the correct perspective on interesting "degenerate" equations in order to be able to introduce innovative tools and methodologies to tackle them.