Aubry sets and the differentiability of the minimal average action in codimension one
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${}^{n}$ $×$ Z, then $u\left(x\right)$ = $u$($x$ + $k$) + $j$ $\phantom{\rule{0.277778em}{0ex}}\forall$$x$. Moser has shown that each of these functions is at finite distance from a plane $u$ = $\rho$ $·$ $x$ and thus has an average slope $\rho$; moreover, Senn has proven that it is possible to define the average action of $u$, which is usually called $\beta \left(\rho \right)$ since it only depends on the slope of $u$. Aubry and Senn have noticed a connection between $\beta \left(\rho \right)$ and the theory of crystals in ${ℝ}^{n+1}$, interpreting $\beta \left(\rho \right)$ as the energy per area of a crystal face normal to $\left(-\rho ,1\right)$. The polar of $\beta$ is usually called -$\alpha$; Senn has shown that $\alpha$ is ${C}^{1}$ and that the dimension of the flat of $\alpha$ which contains $c$ depends only on the “rational space” of ${\alpha }^{\text{'}}$$\left(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.

DOI : https://doi.org/10.1051/cocv:2008017
Classification : 35J20,  35J60
Mots clés : Aubry-Mather theory for elliptic problems, corners of the mean average action
@article{COCV_2009__15_1_1_0,
author = {Bessi, Ugo},
title = {Aubry sets and the differentiability of the minimal average action in codimension one},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1--48},
publisher = {EDP-Sciences},
volume = {15},
number = {1},
year = {2009},
doi = {10.1051/cocv:2008017},
zbl = {1163.35007},
mrnumber = {2488567},
language = {en},
url = {http://www.numdam.org/item/COCV_2009__15_1_1_0/}
}
Bessi, Ugo. Aubry sets and the differentiability of the minimal average action in codimension one. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 1-48. doi : 10.1051/cocv:2008017. http://www.numdam.org/item/COCV_2009__15_1_1_0/

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