Academic year 2011-2012

FROM DISCRETE SYSTEMS TO VARIATIONAL PROBLEMS IN THE CONTINUUM

Scope of the course is to discuss some issues related to a number of problems, ranging from the study of multi-particle systems, to problems in numerical analysis due to the discretization of possibly non convex energies, to the analysis of the atomistic hypothesis of Continuum Mechanics, to the homogenization of discrete structures, and more, and their study through the methods of Gamma-convergence. All these problems have in common the analysis of discrete systems with an increasing number of variables, and have received a lot of attention, from the variational point of view, mainly in the last decade, so as to become a central theme of the research in Applied Mathematics.

Venue: Room Buzano at the Mathematics Department of Politecnico

Tue. October 18 from 3:30 to 5:30 PM

Wed. October 19 from 11 to 13 AM and from 2:30 to 3:30 PM

Thu. October 20 from 10:30 to 12:30 AM

Wed. November 9 from 11 to 13 AM and from 2:30 to 3:30 PM

Thu. November 10 from 10:30 to 12:30 AM

Fri. November 11 from 9:30 to 11:30 AM

Content of the lectures

1. Introduction and motivations from Numerical Analysis, Statistical Mechanics and Continuum Mechanics. Examples of convergence of discrete problems: relaxation and homogenization. Definition and main theorem of Gamma-convergence.

2. Strong/weak convergence of discrete functions as the convergence of their piecewise-constant interpolations. Properties of Gamma-convergence. Lower semicontinuity. Discrete systems with a finite number of parameters. One-point systems. Convergence to a integral funcyional with a piecewise-affine energy density. Nearest-neighbours two-point interactions. Ferromagnetic and anti-ferromagnetic interactions. Convergence to a trivial integral energy. Surface scaling in 1D. Convergence to a phase-transition energy (ferromagnetic case), or an anti-phase transition energy (antiferromagnetic case).

3. Next-to-nearest-neighbours two-point interactions in 1D. Convergence to a phase parameter in the ferro-antiferromagnetic case. Limit energy depending on the jumps and size of the jumps of the phase parameter. Three-point interactions.

4. Ternary systems in 1D: "surfactants". Sets of finite perimeter; compactness, lower semicontinuous surface energies. Binary systems in 2D. Nearest-neighbour interactions: ferromagnetic and antiferromagnetic energies. Anisotropic surface energies.

Next-to-nearest neighbour energies: superposition of surface energies (ferromagnetic case), limits defined on partitions of sets of finite perimeter indexed by order parameters describing ground states. Three-point interactions. Frustration with four-point interactions (square lattice) or nearest-neighbour interaction (triangular lattice).

5. Lennard-Jones interactions. Some motivations from Statistical Mechanics: phase transitions. One-dimensional systems of nearst-neighbour interactions: limits at different scales giving no-tension materials and brittle fracture for rigid materials.

6. "Linearization" of Lennard Jones systems obtaining Griffith brittle fracture with an internal parameter. Next-to-nearest neighbour interactions. Homogenization of the energy density to obtain ground states. Surface relaxation on the fracture set.

7. Some hints at the use of continuous descriptions in the formulation of multiscale problems. Definition of a motion for Lennard-Jones interactions via discrete-in-time approximations.

8. Motion by mean curvature deriving from spin systems: crystalline motion with pinning.