Continuity of the Peierls barrier and robustness of laminations

TL;DR

This paper investigates the continuity properties of the Peierls barrier in monotone variational problems, demonstrating its stability at irrational points and its continuous dependence on parameters, which supports the robustness of laminations of minimizers.

Contribution

It provides a new proof of the Peierls barrier's continuity at irrational points and shows that the existence of laminations is stable under small perturbations in the C1-topology.

Findings

Peierls barrier is continuous at irrational points.

Peierls barrier depends continuously on parameters.

Laminations of minimizers are robust under small C1 perturbations.

Abstract

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.