Asymptotic Analysis

We consider 2D discrete systems, described by scalar functions and governed by periodic interaction potentials. We focus on anisotropic nearest neighbors interactions in the hexagonal lattice and on isotropic long range interactions in the square lattice. In both these cases, we perform a complete Γ-convergence analysis of the energy induced by a configuration of discrete topological singularities. This analysis allows to prove the existence of many metastable configurations of singularities in the hexagonal lattice.

In this paper, one considers the coupling of a Brinkman model and Stokes equations with jump embedded transmission conditions. In this model, one assumes that the viscosity in the porous region is very small. Then we derive a Wentzel–Kramers–Brillouin (WKB) expansion in power series of the square root of this small parameter for the velocity and the pressure which are solution of the transmission problem. This WKB expansion is justified rigorously by proving uniform errors estimates.

We consider the Laplacian in a tubular neighbourhood of a hyperplane subjected to non-self-adjoint PT -symmetric Robin boundary conditions. Its spectrum is found to be purely essential and real for constant boundary conditions. The influence of the perturbation in the boundary conditions on the threshold of the essential spectrum is studied using the Birman–Schwinger principle. Our aim is to derive a sufficient condition for existence, uniqueness and reality of discrete eigenvalues. We show that discrete spectrum exists when the perturbation acts in the mean against the unperturbed boundary conditions and we are able to obtain the first term in its asymptotic expansion in the weak coupling regime.

This paper aims at properly justifying the modeling of a thin piezoelectric plate in unilateral contact with a rigid plane. In order to do that we start from the three-dimensional non-linear Signorini problem which couples the elastic and the electric effects. By an asymptotic analysis we study the convergence of the displacement field and of the electric potential as the thickness of the plate goes to zero. We establish that, at the limit, the in-plane elastic components and the electric potential are coupled and solve a bilateral linear piezoelectric problem. However the transverse mechanical displacement field, is independent of the electric effect and solves a two-dimensional elastic obstacle problem. We also investigate the very popular case of cubic crystals and show that, for thin plates, the piezoelectric coupling effect disappears.

This paper is concerned with boundary blow-up elliptic problems △ u = b ( x ) f ( u ) , x ∈ Ω , u | ∂ Ω = + ∞ , where Ω is a bounded domain with smooth boundary in R N , and b ∈ C loc α ( Ω ) for some α ∈ ( 0 , 1 ) which is nonnegative nontrivial in Ω, but may be vanishing or appropriate singular (including critical singular) on ∂ Ω . Under a new structure condition on f near infinity, we study the exact boundary behavior of such solutions.

We study the homogenization of the Poisson equation in a periodically perforated domain, of period ε > 0 , with a friction type boundary condition on the holes’ boundary. This non-linear condition allows the solution to be non-zero on the periodic boundary if some conditions are satisfied. Using two-scale convergence results we prove that the solution of the mixed variational formulation converges, as ε goes to 0, to the solution of a two-scale mixed problem. We also prove that this homogenized problem is well-posed. A numerical test is done, using the Finite Element Method and a quadratic programming algorithm, in order to compare the heterogeneous and homogenized solutions.

In this paper we study the existence of bound state solutions for stationary Schrödinger systems of the form − Δ u + V ( x ) u = K ( x ) F u ( u , v ) in  R N , − Δ v + V ( x ) v = K ( x ) F v ( u , v ) in  R N , where N ⩾ 3 , V and K are bounded continuous nonnegative functions, and F ( u , v ) is a C 1 and p-homogeneous function with 2 < p < 2 N / ( N − 2 ) . We give a special attention to the case when V may eventually vanishes. Our arguments are based on penalization techniques, variational methods and Moser iteration scheme.