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

Bessi, Ugo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 1-48

Let $ℒ$($x$,$u$,$\nabla$$u$) be a lagrangian periodic of period $1$ in $x_1$,$\cdots$,$x_n$,$u$. We shall study the non self intersecting functions $u$: R${}^n$$\to$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 $u\left(x\right)$ = $u$($x$ + $k$) + $j$ $\forall$$x$. Moser has shown that each of these functions is at finite distance from a plane $u$ = $\rho$ $·$ $x$ and thus has an average slope $\rho$; moreover, Senn has proven that it is possible to define the average action of $u$, which is usually called $\beta \left(\rho \right)$ since it only depends on the slope of $u$. Aubry and Senn have noticed a connection between $\beta \left(\rho \right)$ and the theory of crystals in ${ℝ}^{n+1}$, interpreting $\beta \left(\rho \right)$ as the energy per area of a crystal face normal to $(-\rho ,1)$. The polar of $\beta$ is usually called -$\alpha$; Senn has shown that $\alpha$ is $C^1$ and that the dimension of the flat of $\alpha$ which contains $c$ depends only on the “rational space” of ${\alpha }^{\text{'}}$$(c$). We prove a similar result for the faces (or the faces of the faces, etc.) of the flats of $\alpha$: they are $C^1$ and their dimension depends only on the rational space of their normals.

Classification : 35J20, 35J60
Keywords: Aubry-Mather theory for elliptic problems, corners of the mean average action

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