  We study the Peierls barrier for a broad class of monotone variational
problems. These problems arise naturally in solid state physics and from
Hamiltonian twist maps.
  We start with the case of a fixed local potential and derive an estimate for
the difference of the periodic Peierls barrier and the Peierls barrier of a
general rotation number in a given point. A similar estimate was obtained by
Mather in the context of twist maps, but our proof is different and applies
more generally. It follows from the estimate that the Peierls barrier is
continuous at irrational points.
  Moreover, we show that the Peierls barrier depends continuously on parameters
and hence that the property that a monotone variational problem admits a
lamination of minimizers for a given rotation number, is open in the
C1-topology.

Dynamical Systems