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Aubry sets and the differentiability of the minimal average action in codimension one
Published online by Cambridge University Press: 23 January 2009
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.
- Type
- Research Article
- Information
- ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 1 , January 2009 , pp. 1 - 48
- Copyright
- © EDP Sciences, SMAI, 2008
References
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