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

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 1, page 1-48
- ISSN: 1292-8119

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topBessi, Ugo. "Aubry sets and the differentiability of the minimal average action in codimension one." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 1-48. <http://eudml.org/doc/250567>.

@article{Bessi2009,

abstract = {
Let $\{\cal L\}$(x,u,∇u) be a Lagrangian periodic of period 1 in
x1,...,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∈Rn and (k , j) ∈Zn × 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 = ρ$\cdot$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 $\beta(\rho)$ 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 $\{\bf R\}^\{n+1\}$, interpreting $\beta(\rho)$ as the energy per area of a crystal face normal to $(-\rho,1)$. The polar of β is usually called -α; Senn has shown that α is C1 and that the dimension of the flat of α 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 α: they are C1 and their dimension depends only on the rational space of their normals.
},

author = {Bessi, Ugo},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Aubry-Mather theory for elliptic problems; corners of the mean average action},

language = {eng},

month = {1},

number = {1},

pages = {1-48},

publisher = {EDP Sciences},

title = {Aubry sets and the differentiability of the minimal average action in codimension one},

url = {http://eudml.org/doc/250567},

volume = {15},

year = {2009},

}

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

DA - 2009/1//

PB - EDP Sciences

VL - 15

IS - 1

SP - 1

EP - 48

AB -
Let ${\cal L}$(x,u,∇u) be a Lagrangian periodic of period 1 in
x1,...,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∈Rn and (k , j) ∈Zn × 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 = ρ$\cdot$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 $\beta(\rho)$ 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 ${\bf R}^{n+1}$, interpreting $\beta(\rho)$ as the energy per area of a crystal face normal to $(-\rho,1)$. The polar of β is usually called -α; Senn has shown that α is C1 and that the dimension of the flat of α 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 α: they are C1 and their dimension depends only on the rational space of their normals.

LA - eng

KW - Aubry-Mather theory for elliptic problems; corners of the mean average action

UR - http://eudml.org/doc/250567

ER -

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