Mathieu Lewin et Basile Audoly

Abstract de B. Audoly :  Joint work with Manon Thbaut and Claire Lestringant. When applied to one-dimensional periodic elastic lattices, the method of two-scale expansions produces an elastic energy that depends on both the first and second gradient of the macroscopic displacement u(x), as well as on the scale separation parameter epsilon<<1. By design, two-scale expansions produce solutions u_epsilon(x), that remain smooth in the limit of well separated scales, epsilon->0. In the context of second-order expansions, show that this constrains the higher-order tractions that appear at the boundaries of the equivalent strain-gradient continuum. This leads us to identify a variational principle underpinning second-order homogenization: the macroscopic displacement is shown to be the minimizer of a positive energy functional parameterized by epsilon.

Abstract de M. Lewin : I will present a nonlinear PDE describing the condensate part of an infinite interacting Bose gas. The unknown is a function that does not go to zero at infinity. I will explain that the system undergoes a unique phase transition, with the solution going from the constant function at low density (describing a superfluid) to a wave oscillating everywhere in space at high density (describing a supersolid). Numerical simulations suggest that the latter is periodic. Work in collaboration with Nam (Munich).