21 nov 2012 -- 17:00   [open in google calendar]

Sala Seminari, Department of Mathematics, Pisa University

Abstract.

We consider a steepest descent flow for the modified total squared curvature defined on curves. We call the flow the shortening-straightening flow. First it has been proved by A. Polden (1996) that the flow admits smooth solutions globally defined in time, when the initial curve is smooth, closed, and has finite length. In 2002, G. Dziuk, E. Kuwert, and R. Schätzle extended Polden's result to closed curves with finite length in n-dimensional Euclidean space. We are interested in the following problem: ``What is a dynamics of non-closed planar curves with infinite length governed by shortening-straightening flow?" In this talk, we will talk about a long time existence of a solution of the shortening-straightening flow starting from a non-closed planar curve with infinite length. Moreover we show that the solution converges to a stationary solution as time goes to infinity. This work is a joint research with M. Novaga (Padova Univ.).