Mathematics

In these lectures we consider curves evolving in the plane by the gradient flow of some geometric functionals, with particular attention to the problem of regularity. In a first part we discuss briefly the motion of a regular planar curve by mean curvature and we will analyze the cases where smoothness is preserved during the flow. Then we will study the works of Polden about evolution of curves by the gradient of functionals involving the curvature: we will discuss regularity, convergence and classification of the limit curves. If there will be time, we will present some extensions of these results, applications, open problems and research directions.