Seminar

The level set method has been used with great success the last thirty years in both pure and applied mathematics to describe evolutions of various physical situations. In mean curvature flow, the evolving hyper surface (front) is thought of as the level set of a function that satisfies a nonlinear degenerate parabolic equation. Solutions have always been defined in the viscosity sense. Viscosity solutions are functions that in general may not even be differentiable (let alone twice differentiable) but satisfy a second order differential equation in a weak sense. For a monotonically advancing front, I will describe why viscosity solutions are in fact twice differentiable and satisfy the equation in the classical sense. The proof weaves together analysis and geometry.