Abstract : The celebrated mean-curvature flow describes the evolution of the interface between two domains which moves so that its orthogonal velocity at each point is proportional to its mean curvature, pointing in the direction of decreasing the curvature. In the second order mean-curvature flow, it is the derivative of the orthogonal velocity (i.e., the acceleration) that is proportional to the mean curvature. Both flows can be described through partial differential equations (PDEs) for the associated arrival time functions. However, unlike the PDE for the classical mean-curvature flow, the equation for its second order version -- which we refer to as the ``cascade equation" -- is hyperbolic and does not enjoy the comparison principle. For this reason, and due to other challenges, the standard PDE tools are not sufficient to develop a well-posedness theory for the cascade equation directly. Nevertheless, it turns out that solutions to the cascade PDE can be identified with minimal elements of a set of value functions in a family of mean-field games. As a result, the existence of a solution to the cascade equation can be shown by proving the compactness of the aforementioned set of value functions, which we accomplish by employing the tools from Geometric Measure Theory. Joint work with Y. Guo and M. Shkolnikov.

Keywords : mean-curvature flow; cascade equation; mean-field games