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

Ugo Bessi

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

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

Abstract

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Let 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 → R minimizing 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 ∀ 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 𝐑 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 C1 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 C1 and their dimension depends only on the rational space of their normals.

How to cite

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Bessi, 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 -

References

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