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

Bessi, Ugo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 1, pp. 1-48.

Let $ℒ$($x$,$u$,$\nabla$$u$) be a lagrangian periodic of period $1$ in $x_1$,$\cdots$,$x_n$,$u$. We shall study the non self intersecting functions $u$: R${}^n$$\to$R minimizing $ℒ$; non self intersecting means that, if $u$($x_0$ + $k$) + $j$ = $u$($x_0$) for some $x_0$ $\in$ R${}^n$ and ($k$ , $j$) $\in$ Z${}^n$ $\times$ Z, then $u\left(x\right)$ = $u$($x$ + $k$) + $j$ $\forall$$x$. Moser has shown that each of these functions is at finite distance from a plane $u$ = $\rho$ $·$ $x$ and thus has an average slope $\rho$; moreover, Senn has proven that it is possible to define the average action of $u$, which is usually called $\beta \left(\rho \right)$ since it only depends on the slope of $u$. Aubry and Senn have noticed a connection between $\beta \left(\rho \right)$ and the theory of crystals in ${ℝ}^{n+1}$, interpreting $\beta \left(\rho \right)$ as the energy per area of a crystal face normal to $(-\rho ,1)$. The polar of $\beta$ is usually called -$\alpha$; Senn has shown that $\alpha$ is $C^1$ and that the dimension of the flat of $\alpha$ which contains $c$ depends only on the “rational space” of ${\alpha }^{\text{'}}$$(c$). We prove a similar result for the faces (or the faces of the faces, etc.) of the flats of $\alpha$: they are $C^1$ and their dimension depends only on the rational space of their normals.

[1] G. Alberti, L. Ambrosio and X. Cabré, On a long standing conjecture of De Giorgi: symmetry in $3$d for general nonlinearities and a local minimality property. Acta Appl. Math. 65 (2001) 9-33. | MR | Zbl

[2] S. Aubry and P.Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. Physica 8D (1983) 381-422.

[3] F. Auer and V. Bangert, Differentiability of the stable norm in codimension one. CRAS 333 (2001) 1095-1100. | MR | Zbl

[4] V. Bangert, On minimal laminations of the torus. Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 95-138. | Numdam | MR | Zbl

[5] V. Bangert, Geodesic rays, Busemann functions and monotone twist maps. Calc. Var. 2 (1994) 49-63. | MR | Zbl

[6] P. Bernard and B. Buffoni, Optimal mass transportation and Mather theory. J. Eur. Math. Soc. 9 (2007) 85-121. | MR

[7] D. Burago, S. Ivanov and B. Kleiner, On the structure of the stable norm of periodic metrics. Math. Res. Lett. 4 (1997) 791-808. | MR | Zbl

[8] L. De Pascale, M.S. Gelli and L. Granieri, Minimal measures, one-dimensional currents and the Monge-Kantorovich probem. Calc. Var. Partial Differential Equations 27 (2006) 1-23. | MR | Zbl

[9] K. Deimling, Nonlinear Functional Analysis. Springer, Berlin (1985). | MR | Zbl

[10] M.P. Do Carmo, Differential Forms and Applications. Springer, Berlin (1994). | MR | Zbl

[11] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers. Oxford (1980). | MR | Zbl

[12] D. Massart, Stable norms of surfaces: local structure of the unit ball at rational directions. GAFA 7 (1997) 996-1010. | MR | Zbl

[13] D. Massart, On Aubry sets and Mather's action functional. Israel J. Math. 134 (2003) 157-171. | MR | Zbl

[14] J.N. Mather, Differentiability of the minimal average action as a function of the rotation number. Bol. Soc. Bras. Mat. 21 (1990) 59-70. | MR | Zbl

[15] J.N. Mather, Action minimizing invariant measures for positive-definite Lagrangian systems. Math. Zeit. 207 (1991) 169-207. | MR | Zbl

[16] J.N. Mather, Variational construction of connecting orbits. Ann. Inst. Fourier 43 (1993) 1349-1386. | Numdam | MR | Zbl

[17] J. Moser, Minimal solutions of variational problems on a torus. Ann. Inst. H. Poincaré Anal. Non LinÈaire 3 (1989) 229-272. | Numdam | MR | Zbl

[18] O. Osuna, Vertices of Mather's beta function. Ergodic Theory Dynam. Systems 25 (2005) 949-955. | MR | Zbl

[19] P.H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation. Comm. Pure Appl. Math. 56 (2003) 1078-1134. | MR

[20] W. Senn, Strikte Konvexität für Variationsprobleme auf dem n-dimensionalen Torus. Manuscripta Math. 71 (1991) 45-65. | MR | Zbl

[21] W. Senn, Differentiability properties of the minimal average action. Calc. Var. Partial Differential Equations 3 (1995) 343-384. | MR | Zbl

[22] W. Senn, Equilibrium form of crystals and the stable norm. Z. angew. Math. Phys. 49 (1998) 919-933. | MR | Zbl

[23] J.E. Taylor, Crystalline variational problems. BAMS 84 (1978) 568-588. | MR | Zbl

[24] M.E. Taylor, Partial Differential Equations, Basic Theory Springer, Berlin (1996). | MR | Zbl

[25] N. Wiener, The ergodic theorem. Duke Math. J 5 (1939) 1-18. | JFM | MR