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

Dipartimento di Matematica,
Università Roma Tre, Largo S.
Leonardo Murialdo, 00146 Roma, Italy. bessi@matrm3.mat.uniroma3.it

Received:
3
November
2006

Revised:
23
July
2007

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.

Mathematics Subject Classification: 35J20 / 35J60

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

*© EDP Sciences, SMAI, 2008*