Abstract
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.
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Mantegazza, C., Novaga, M., Tortorelli, V.M. (2004). Evolution by Curvature of Networks of Curves in the Plane. In: Baird, P., Fardoun, A., Regbaoui, R., El Soufi, A. (eds) Variational Problems in Riemannian Geometry. Progress in Nonlinear Differential Equations and Their Applications, vol 59. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7968-2_7
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DOI: https://doi.org/10.1007/978-3-0348-7968-2_7
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