Highly degenerate harmonic mean curvature flow

In this thesis we consider a fully nonlinear parabolic evolution equation for a hypersurface in the Euclidean space. We study the existence of solutions to the Harmonic Mean Curvature Flow for weakly convex surfaces. We prove short time existence for weakly convex surfaces with flat sides as well as optimal regularity. We also show that the boundaries of the flat sides evolve by the Curve Shortening Flow. Our results follow by using the Inverse Function Theorem in suitable Banach Spaces scaled according to the degenerate problem. We establish sharp a-priori estimates for the linearized problem in these spaces.