Aubry sets and the differentiability of minimal average action in codimension one

Let $\L(x,u,\nabla u)$ be a Lagrangian periodic of period $1$ in $x_1,\dots,x_n,u$. We shall study the non self intersecting functions $\fun{u}{\R^n}{\R}$ minimizing $\L$; non self intersecting means that, if $u(x_0+k)+j=u(x_0)$ for some $x_0\in\R^n$ and $(k,j)\in\Z^n\times\Z$, then $u(x)= u(x+k)+j\forall x$. J. Moser has shown that each of these functions is at finite distance from a plane $u=\r\cdot x$ and thus has an average slope $\r$; moreover, W. Senn has proven that it is possible to define the average action of $u$, which is usually called $\b(\r)$ since it only depends on the slope of $u$. Aubry and Senn have noticed a connection between $\b(\r)$ and the theory of crystals in $\R^{n+1}$, interpreting $\b(\r)$ as the energy per area of a crystal face normal to $(-\r,1)$. The polar of $\b$ is usually called $-\a$; Senn has shown that $\a$ is $C^1$ and that the dimension of the flat of $\a$ which contains $c$ depends only on the ``rational space'' of $\ader$. We prove a similar result for the faces (or the faces of the faces, etc.) of the flats of $\a$: they are $C^1$ and their dimension depends only on the rational space of their normals.