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We study the Peierls barrier for a broad class of monotone variationalproblems. These problems arise naturally in solid state physics and fromHamiltonian twist maps.  We start with the case of a fixed local potential and derive an estimate forthe difference of the periodic Peierls barrier and the Peierls barrier of ageneral rotation number in a given point. A similar estimate was obtained byMather in the context of twist maps, but our proof is different and appliesmore generally. It follows from the estimate that the Peierls barrier iscontinuous at irrational points.  Moreover, we show that the Peierls barrier depends continuously on parametersand hence that the property that a monotone variational problem admits alamination of minimizers for a given rotation number, is open in theC1-topology.

Continuity of the Peierls barrier and robustness of laminations