Existence of Periodic Solutions for Equations of Evolving Curves

Abstract

Evolution equations of curvatures of convex curves are considered by the Gauss map parametrization. A time periodic unstable solution is constructed for a reasonable class of time periodic data. Our solution is arranged to satisfy a constraint so that it yields closed, embedded, convex curves moving periodically in time (up to translation) whose normal speed equals the curvature minus a given time periodic function depending on curves only through its normals. For curvatures of periodically evolving curves a priori lower and upper bounds depending only on periodic data are obtained. A new penalty method is introduced so that our solution satisfies the constraint. Solutions of penalized equations are constructed by adapting the degree theory.