WG4 Research Seminar - Talk by Anna Dall'Acqua

On January 19th, 2022, Anna Dall'Acqua (University of Ulm) will give a talk at 2:30pm CET about

"On the elastic flow of networks".

The talk will be held online in Zoom as part of the research seminar Analysis of the FernUniversität in Hagen. The zoom URL will be sent to all members of the WG4 and to all further colleagues who send a mail to Delio Mugnolo.

Abstract

In the Bernoulli model of an elastic rod described by a curve, the elastic energy is given by the integral over the curve of the curvature squared. Due to the behavior of the energy under scaling, one soon notices that the length of the curve should be penalized (or fixed) to find minimizers.

For this penalized energy we consider the steepest descent flow. That is, a one-parameter family of curves that changes in time in such a way that the energy decreases as fast as possible, i.e., in the direction of the L2-gradient of the energy. This gives a fourth-order geometric flow. For closed curves, it is well known that starting from a smooth initial datum the solution exists for all time and sub-converges to a critical point.

We extend this result to networks of 3-curves. We consider networks of 3-curves in Rn, where each of the three curves is fixed at one end-point and at the other is joint to the other curves at a movable 3-junction, a three-pointed star network. For this geometric evolution problem with natural boundary conditions, we provide a longtime existence result and a sub-convergence result under some mild topological assumptions.

This is joint work with Chun-Chi Lin (National Taiwan Normal University) and Paola Pozzi (University of Duisburg-Essen).