Let $\mathcal{L}$($x$,$u$,$\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 $\mathcal{L}$; 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 $u\left(x\right)$ = $u$($x$ + $k$) + $j$ $\phantom{\rule{0.277778em}{0ex}}\forall $$x$. Moser has shown that each of these functions is at finite distance from a plane $u$ = $\rho $ $\xb7$ $x$ and thus has an average slope $\rho $; moreover, Senn has proven that it is possible to define the average action of $u$, which is usually called $\beta \left(\rho \right)$ since it only depends on the slope of $u$. Aubry and Senn have noticed a connection between $\beta \left(\rho \right)$ and the theory of crystals in ${\mathbb{R}}^{n+1}$, interpreting $\beta \left(\rho \right)$ as the energy per area of a crystal face normal to $(-\rho ,1)$. The polar of $\beta $ is usually called -$\alpha $; Senn has shown that $\alpha $ is ${C}^{1}$ and that the dimension of the flat of $\alpha $ which contains $c$ depends only on the “rational space” of ${\alpha}^{\text{'}}$$(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.

Keywords: 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}, mrnumber = {2488567}, zbl = {1163.35007}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008017/} }

TY - JOUR AU - Bessi, Ugo TI - Aubry sets and the differentiability of the minimal average action in codimension one JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2009 SP - 1 EP - 48 VL - 15 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008017/ DO - 10.1051/cocv:2008017 LA - en ID - COCV_2009__15_1_1_0 ER -

%0 Journal Article %A Bessi, Ugo %T Aubry sets and the differentiability of the minimal average action in codimension one %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 1-48 %V 15 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008017/ %R 10.1051/cocv:2008017 %G en %F 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, Volume 15 (2009) no. 1, pp. 1-48. doi : 10.1051/cocv:2008017. http://archive.numdam.org/articles/10.1051/cocv:2008017/

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