Homogenization of geometric flows on lattices
speaker: Andrea Braides (Università degli Studi di Roma "Tor Vergata")abstract: In this talk I will touch two subjects that had attracted the attention of De Giorgi in his last years. The fi st is the study of energies depending on discrete quantities, in particular on fi diff ences. Indeed, one of his last proposed conjectures, proved by Gobbino, concerns some energies directly inspired by finite-difference schemes in Image Reconstruction set in a variational framework by Chambolle. The other subject is the theory of minimizing movements, or implicit Euler schemes, which is at the basis of a modern theory of gradient fl ws as presented in the now classical book by Ambrosio, Gigli and Savar. De Giorgi was aware that the application of the minimizing-movement scheme to non- convex oscillating energies may present unforeseen difficulties in the correct description of a homogenized motion, due to the complex effect of local minimization. The use of discrete energies, in particular of those defined on a lattice, has allowed on one hand to control with more ease the effect of local minimization, and on the other hand the introduction of a framework where one may defined crystalline surface energies in a discrete-to-continuum process. Such energies are related to motion by crystalline curvature, which can be sometimes described as a simple system of ODEs. I will present the general framework of minimizing movements along a sequence of energies at a given time scale, and provide some connection with the corresponding Gamma-limit. I will then specialize that framework to spin systems, remarking that we always have pinning (no motion) for fast time scales, and motion by crystalline curvature for slow time scales. At one or more critical scales the motion is influenced in various ways by local minimization and require a macroscopic description depending on the interaction by space and time scales
timetable:
Wed 21 Sep, 11:45 - 12:40, Palazzo dei Congressi
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