Dipartimento di Matematica, Università Roma Tre, Largo S. Leonardo Murialdo, 00146 Roma, Italy. email@example.com
Let (x,u,∇ u) be a Lagrangian periodic of period 1 in x 1,...,x n ,u. We shall study the non self intersecting functions u: R n R minimizing ; non self intersecting means that, if u(x 0 + k) + j = u(x 0) for some x 0 ∈ R n and (k , j) ∈ Z n × Z, then u(x) = u(x + k) + j x. Moser has shown that each of these functions is at finite distance from a plane u = ρ x and thus has an average slope ρ; moreover, Senn has proven that it is possible to define the average action of u, which is usually called since it only depends on the slope of u. Aubry and Senn have noticed a connection between and the theory of crystals in , interpreting as the energy per area of a crystal face normal to . The polar of β is usually called -α; Senn has shown that α is C 1 and that the dimension of the flat of α which contains c depends only on the “rational space” of (c). We prove a similar result for the faces (or the faces of the faces, etc.) of the flats of α: they are C 1 and their dimension depends only on the rational space of their normals.
(Received November 3 2006)
(Revised July 23 2007)
(Online publication January 23 2009)
Mathematics Subject Classification: