Let ${\cal L}$(x,u,∇u) be a Lagrangian periodic of period 1 in
x1,...,xn,u. We shall study the non self intersecting
functions u: Rn${\to}$R minimizing ${\cal L}$; non self intersecting means that, if u(x0 + k) + j = u(x0)
for some x0∈Rn and (k , j) ∈Zn × Z, then
u(x) = u(x + k) + j$\;\forall$x. Moser has shown that each of these
functions is at finite distance from a plane
u = ρ$\cdot$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 $\beta(\rho)$ since it only depends on the slope of u.
Aubry and Senn have noticed a connection between $\beta(\rho)$ and the
theory of crystals in ${\bf R}^{n+1}$, interpreting $\beta(\rho)$ as the energy per area of a crystal face normal to $(-\rho,1)$. The polar of β is usually called -α; Senn has shown that α is C1 and that the dimension of the flat of α which contains c depends only on the “rational space” of $\alpha^\prime$(c). We prove a similar result for the faces (or the faces of the faces, etc.) of the flats of α: they are C1 and their dimension depends only on the rational space of their normals.