Let L(x, u, ∇ u) be a Lagrangian periodic of period 1 in x1, …, xn, u. J. Moser has shown that, among the minimizers of L, the non self intersecting ones (i. e. if u(x0 + k) + j = u(x0) for some x0 ∈ Rn and (k,j) ∈ Zn × Z, then u ≡ u(x+k)+j) enjoy some particularly simple properties. In particular, each of them 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 an average action which only depends on ρ; this function is usually called b(ρ). Aubry and Senn have noticed that b(ρ) can be interpreted as the energy per area of the face of a crystal in Rn+1; the normal to this face is (-ρ,1)⁄√(|ρ|2+1). The polar of b is usually called -a; W. Senn has shown that a is C1 and that the dimension of the flat of a which contains c depends only on the “rational space” of a'(c). We prove a similar result for the faces (or the faces of the faces, etc.) of the flats of a: they are C1 and and their dimension depends only on the rational space of their normals.
We will present a joint work with C. Le Bris and P.-L. Lions dealing with the definition of the energy of infinite set of atoms. Starting from the very simple case of a periodic lattice with classical energy, we will extend this kind of results to stochastic lattices and/or quantum models of Thomas-Fermi type.
Periodic and quasi-periodic orbits of the n-body problem are critical points of the action functional constrained to the Sobolev space of symmetric loops. Variational methods yield collisionless orbits provided the group of symmetries fulfills certain conditions (such as the rotating circle property). Here we show how to constructively classify all symmetry groups satisfying such hypothesis and illustrate which minima can occur.
We consider two-dimensional Schrödinger operators in bounded domains. We analyze the relations between the nodal domains, the spectral minimal partitions and the spectral properties of the corresponding operators.
We consider some generalizations of the Aubry-Mather theory for elliptic (possibly singular or degenerate) PDEs, which are a model for phase transitions and fluid jets. We construct solutions whose level sets lie at bounded distance from any given hyperplane. Such solutions may be seen as an extension of the trajectories with prescribed rotation number in the dynamical system framework. When the PDE does not depend on the space variable, the problem is related to a conjecture of De Giorgi. Multiplicity results and applications to statistical mechanics will be also discussed.