**Authors**
: Alessandro Giuliani, Joel L. Lebowitz and Elliott H. Lieb

**Title: Periodic minimizers in 1D local mean field theory
**

**Abstract**:
Using reflection positivity techniques we prove the
existence of minimizers for a class of mesoscopic free-energies
representing 1D systems with competing interactions.
All minimizers are either periodic, with zero average, or of
constant sign. If the local term in the free energy satisfies a convexity
condition, then all minimizers are either periodic or constant.
Examples of both phenomena are given. This extends our previous
work where such results were proved for the ground states of lattice
systems with ferromagnetic nearest neighbor interactions and dipolar
type antiferromagnetic long range interactions.

**Keywords**:
Free energy functionals; periodic minimizers; reflection positivity.

Alessandro Giuliani,

Dipartimento di Matematica,
Universita' di Roma Tre,
L.go S. Leonardo Murialdo 1, 00146 Roma, Italy

giuliani@mat.uniroma3.it

Joel L. Lebowitz,

Department of Mathematics and Physics, Rutgers University,
Piscataway, NJ 08854 USA.

lebowitz@math.rutgers.edu

Elliott H. Lieb,

Department of Mathematics and Physics,
Princeton University, Princeton 08544 NJ, USA

lieb@princeton.edu