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 (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 β(ρ) since it only depends on the slope of u. Aubry and Senn have noticed a connection between β(ρ) and the theory of crystals in n+1 , interpreting β(ρ) as the energy per area of a crystal face normal to (-ρ,1). 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 α ' (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.

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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