On the Classification of Networks Self–Similarly Moving by Curvature

We give an overview of the classification of networks in the plane with at most two triple junctions with the property that under the motion by curvature they are self-similarly shrinking. After the contributions in [7, 9, 20], such a classification was completed in the recent work in [4] (see also [3]), proving that there are no self-shrinking networks homeomorphic to the Greek “theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees. We present the main geometric ideas behind the work [4]. We also briefly introduce our work in progress in the higher-dimensional case of networks of surfaces in R

Motion by curvature of networks with two triple junctions

We consider the evolution by curvature of a general embedded network with two triple junctions. We classify the possible singularities and we discuss the long time existence of the evolution.

Computer Simulations Combining Finite Difference and Finite Element Methods: Solute Drag on Migrating Grain Boundaries in Three-Dimension

The current study aims to improve our fundamental understanding of solute segregation and solute drag on migrating grain boundaries (GB) in three dimensions. Computer simulation combines finite difference and finite element methods. An exemplary case study is reported, in which a spherical grain is embedded inside a cubic grain and shrinks as a result of motion by curvature, as a preliminary to modeling grain growth in single phase materials. The results agree qualitatively with literature studies in 1-D.