Information   Participants   Program

This is an informal workshop bringing together mathematicians and mathematical physicists from Athens, L’Aquila, and Rome.

A tentative list of topics includes:

 •  Connections between stochastic and deterministic models of phase transitions
 •  Elliptic problems with variational structure

The workshop is organized by Vassilis Papanicolaou and is partially supported by a NTUA ΠEBE 2008 grant. Any interested participants should contact the organizer at .

Participants

  Nicholas Alikakos (Athens)
  Michael Berketis (Athens)
  Apostolos Damialis (Athens)
  Anna De Masi (L’Aquila)
  Giorgio Fusco (L’Aquila)
  Nikos Karachalios (Karlovassi)
  Georgia Karali (Heraklion)
  Nikolaos Katzourakis (Athens)
  Michael Nikolouzos (Athens)
  Vassilis Papanicolaou (Athens)
  Cristina Pignotti (L’Aquila)
  Errico Presutti (Rome)
  Dimitrios Tsagkarogiannis (Rome)

Program

Anna De Masi
Some open problems in the theory of open systems
We consider states where a current flows in the system (as, for instance, when heat flows through a conductor whose sides are kept at different temperatures). The macroscopic theory of such systems is well understood if there is no phase transition, in other words, when the energy functional is strictly convex. In this case, the  analysis of some stochastic particle models leads to interesting problems at the macroscopic level where fluctuations can be incorporated in the theory by introducing suitable penalty functionals and corresponding variational problems. We would like to consider the case of nonconvex free energy.

Giorgio Fusco
We are interested in the existence of solutions u : Rⁿ → Rⁿ for Δu = W_u (u), x in Rⁿ, equivariant with respect to a discrete reflection group G of infinite order. In this case, u has spatial periodic structure (lattice or crystallographic structure). We also discuss the existence of solutions for the Dirichlet problem for the equation on a general bounded domain. In the first problem, in the case where u is scalar, it has been proved that the equation possesses ‘saddle solutions’, that is, solutions that vanish on two orthogonal lines. It has been conjectured (Gui, 2007) that θ = ^π⁄₂ is the only possibility. We would like to discuss why, instead, it is reasonable to expect that saddle solutions with θ ≠ ^π⁄₂ do exist and the difficulties for a rigorous proof.

Errico Presutti
The problem of existence of interfaces for functionals arising from the Potts model
The mesoscopic version of the Potts model with n ≥ 3 species is described by a functional of the Allen–Cahn type. It is known that the absolute minimizers are spatially constant functions. For a critical value of a parameter (whose physical meaning is the temperature), there are n +1 minimizers, n of them are related to each other by a symmetry transformation and are called ‘ordered phases’, the (n +1)-th minimizer being called the ‘disordered phase’. Open problems of physical interest concern the existence of spatially nonhomogeneous stationary solutions which asymptotically, away from the origin, have the n ordered phases. Based on the analysis of the Potts model on the lattice with nearest neighbor interactions, the conjecture is that such a solution may not exist.

Dimitrios Tsagkarogiannis
Ground states of a magnetic system with short and long range interactions
Magnetic systems with short range ferromagnetic and long range antiferromagnetic interactions at zero temperature exhibit periodic patterns on a scale determined by the competition between the interactions. Moreover, in the presence of an increasing external magnetic field, it has been conjectured the transition from a periodic structure with ‘stripes’ to a pattern with ‘balls’. We investigate these issues for a mesoscopic functional with a short range part being the intefacial energy and a long range part being some nonlocal interaction energy.

Last modified on 18 November 2008.