Dottorato di ricerca in Matematica
per le Scienze dell'Ingegneria - Politecnico di Torino
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
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
convergence of discrete problems: relaxation and homogenization.
Definition and main theorem of
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
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.