Asymptotic Analysis

We analyze a general class of difference operators H_ε=T_ε+V_ε on ℓ²((εZ)^d), where V_ε is a one-well potential and ε is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of H_ε. These are obtained from eigenfunctions or quasimodes for the operator H_ε, acting on L²(R^d), via restriction to the lattice (εZ)^d.

We use variational methods to study the asymptotic behavior of solutions of p-Laplacian problems with nearly subcritical non-linearity in general, possibly non-smooth, bounded domains.

We study non-linear ground states of the Gross–Pitaevskii equation in the space of one, two and three dimensions with a radially symmetric harmonic potential. The Thomas–Fermi approximation of ground states on various spatial scales was recently justified using variational methods. We justify here the Thomas–Fermi approximation on an uniform spatial scale using the Painlevé-II equation. In the space of one dimension, these results allow us to characterize the distribution of eigenvalues in the point spectrum of the Schrödinger operator associated with the non-linear ground state.

In this paper we discuss the concept of stochastic two-scale convergence, which is appropriate to solve coupled-periodic and stochastic homogenization problems. This concept is a combination of both well-known two-scale convergence and stochastic two-scale convergence in the mean schemes, and is a generalization of the said previous methods. By way of illustration we apply it to solve a homogenization problem related to an integral functional with convex integrand. This problematic relies on the notion of dynamical system which is our basic tool.