Calculus of Variations and Geometric Measure Theory

G. Scilla

Variational motion of discrete interfaces

created by scilla on 22 Nov 2013
modified on 11 Oct 2017


Phd Thesis

Inserted: 22 nov 2013
Last Updated: 11 oct 2017

Year: 2014

PhD Thesis (updated version) - Supervisor: Prof. Andrea Braides - Thesis defended on 09 January 2014 in front of a Board of Examiners composed by:

Prof. Etienne Sandier (chairman)

Prof. Marco Cicalese

Prof. Matteo Novaga

Links: Published on PADIS uniroma1


In this thesis we study evolution of physical systems driven by interfacial type energies in presence of dissipation, by coupling the minimizing movements scheme for geometric evolutions due to Almgren, Taylor and Wang and a discrete-to-continuum analysis via $\Gamma$-convergence. This new approach has been recently introduced by Braides, Gelli and Novaga (see Chapter 4) to study the motion of discrete interfaces of nearest neighbors interacting ferromagnetic systems.

After some introductory material, in Chapter 5 we study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional periodic ``high-contrast'' environment (that is, with periodic inclusions not energetically favorable). We show (with respect to the case of no inclusions) that, in general, the effective limit motion does not depend only on the $\Gamma$-limit, but also on geometrical features that are not detected in the static description. In particular, we show how the pinning threshold is influenced by the microstructure and that the effective motion is described by a new homogenized velocity, independent of the value of the inclusion. In Chapter 6 we treat the case of a “low-contrast” periodic medium, and we show that, below a threshold value of the contrast parameter, the effective velocity function and the pinning threshold depend on the value of the contrast parameter; above this threshold, instead, the motion is constrained as in the high-contrast case. Chapter 7 deals with geometric ``backward'' motions, i.e. reversed time motions, by coupling, as before, the minimizing movements approach and a discrete-to-continuum analysis. This type of motions is ill-posed in the continuous case and it can be described in the discrete one by a scaling-in-time argument. Starting from a point, the discrete motion is a family of checkerboard sets increasing by inclusion and shape preserving if the nucleus contains the initial point. The limit motion is a family of nucleating and linearly expanding sets whose shape depends on the chosen metric. In view of the possible definition of motion in random media and motivated by finding at least an estimate for the pinning threshold, in Appendix we treat homogenization of energies associated to a "rigid" random spin system by using percolation techniques.

Keywords: discrete systems, minimizing movements, flat flows, motion by curvature, crystalline curvature, Geometric motion, Backward motion, Nucleation, periodic media, low contrast media, percolation, Geometric minimizing movements, curvature-driven motion