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ESAIM: COCV
DOI: 10.1051/cocv:2008017

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

Ugo Bessi

Dipartimento di Matematica, Università Roma Tre, Largo S. Leonardo Murialdo, 00146 Roma, Italy


(Received November 3, 2006. Revised July 23, 2007. Published online March 6, 2008.)

Abstract
Let ${\cal L}$(x,u,$\nabla$u) be a Lagrangian periodic of period 1 in x1,$\dots$,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 $\in$ Rn and (k , j) $\in$ Zn $\times$ 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 = $\rho$ $\cdot$ 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 $\beta(\rho)$ 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 Rn+1, interpreting $\beta(\rho)$ 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 C1 and that the dimension of the flat of $\alpha$ 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 $\alpha$: they are C1 and their dimension depends only on the rational space of their normals.


Mathematics Subject Classification. 35J20, 35J60

Key words: Aubry-Mather theory for elliptic problems, corners of the mean average action


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