Calculus of Variations and Geometric Measure Theory

C. Mantegazza

Evolution by Curvature of Networks of Curves in the Plane

created on 07 Apr 2004
modified on 04 Jun 2013


Published Paper

Inserted: 7 apr 2004
Last Updated: 4 jun 2013

Journal: Progress in Nonlin. Diff. Eqs. and Appl.
Volume: 59
Pages: 95-109
Year: 2004


This survey describes our project to study the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the Length functional.

Such a flow can represent the evolution of a two--dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries.

Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which are ``essentially'' non regular.

In this paper, we introduce the problem and we present some results and open problems about existence, uniqueness and, in particular, the global regularity of the flow.