Texas A&M
University, Department of
Mathematics, 216 Milner Hall, 15th of April 2009, 3:00-3:50
Groups and Dynamics Seminar
Equilibrium solutions for some variational problems with group symmetries
Enrico Valdinoci of Seconda Universita di Roma "Tor Vergata", Italy
Motivated by statistical mechanics we consider a set of "atoms" whose
state is characterized by a number. The atoms interact with each other,
but the interactions can be long range or depend on the state of
several atoms.
We assume (roughly) that 1) The interaction is invariant under
the action of a group G and with respect to addition of an integer to
the configuration of all sites. The group G is "residually
finite" (The most famous example is when G is an integer lattice) and
the action has a finite fundamental domain. 2) The interaction favors
alignment (ferromagnetism) 3) The interaction decreases fast enough, is
coercive, and moderately smooth.
We show that:
A) For every cocycle $\phi$ of the group G there is a minimizing
configuration $x$ such that $|x(g i) - x(i) - \phi(g) | \le M$.
B) If there is not a continuous family of minimizers with the above
property, there is an equilibrium configuration corresponding to the
same cocycle
which is not a minimizer.
Notes:
When the group G is the integers, the interactions are nearest
neighbor, the above results were proved by A) Aubry-LeDaeron and Mather
B)Mather.
There are generalizations to Partial differential equations and to other continuum models.
The above is joint work with R. de la Llave.