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

Issue		ESAIM: COCV Volume 15, Number 1, January-March 2009


Page(s)		1 - 48
DOI		10.1051/cocv:2008017
Published online		23 January 2009

ESAIM: COCV 15 (2009) 1-48
DOI: 10.1051/cocv:2008017

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

Ugo Bessi

Dipartimento di Matematica, Università Roma Tre, Largo S. Leonardo Murialdo, 00146 Roma, Italy. bessi@matrm3.mat.uniroma3.it

Received November 3, 2006. Revised July 23, 2007. Published online March 6, 2008.

Abstract
Let ${\cal L}$ (x,u, $\nabla$ u) be a Lagrangian periodic of period 1 in x₁, $\dots$ ,x_n,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 $\in$ Rⁿ and (k , j) $\in$ Zⁿ $\times$ 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 = $\rho$ $\cdot$ 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 $\beta(\rho)$ 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 $\beta$ is usually called - $\alpha$ ; Senn has shown that $\alpha$ is C¹ and that the dimension of the flat of $\alpha$ 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 $\alpha$ : they are C¹ and their dimension depends only on the rational space of their normals.

Mathematics Subject Classification. 35J20, 35J60

Key words: Aubry-Mather theory for elliptic problems, corners of the mean average action

© EDP Sciences, SMAI 2008

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