Volume 15, Number 1, January-March 2009
|Page(s)||1 - 48|
|Published online||23 January 2009|
Aubry sets and the differentiability of the minimal average action in codimension one
Dipartimento di Matematica,
Università Roma Tre, Largo S.
Leonardo Murialdo, 00146 Roma, Italy. firstname.lastname@example.org
Revised: 23 July 2007
Let (x,u,∇u) be a Lagrangian periodic of period 1 in x1,...,xn,u. We shall study the non self intersecting functions u: RnR minimizing ; non self intersecting means that, if u(x0 + k) + j = u(x0) for some x0 ∈ 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 since it only depends on the slope of u. Aubry and Senn have noticed a connection between and the theory of crystals in , interpreting as the energy per area of a crystal face normal to . 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 (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.
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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