Anne Beaulieu

A Ginzburg–Landau problem with weight having minima on the boundary

In this paper, we study the following Ginzburg-Landau functional: E(epsilon)(u) = 1/2 integral(G) p\del u\(2) + 1/4 epsilon(2) integral(G) p(1-\u\(2))(2), where u is an element of H(g)(1) (G, C), and p is a smooth bounded and non-negative map, having minima on the boundary of (G) over bar. We give the location of the singularities in the case where the degree around each singularity is equal to 1.

Some remarks on the linearized operator about the radial solution for the Ginzburg–Landau equation

Abstract We consider the linearized operators, denoted L d,1 , of the Ginzburg–Landau operator Δu+u(1−|u|2) in R2, about the radial solutions ud,1(x)=fd(r)eidθ, for all d⩾1. We state the correspondence between the real vector space of the bounded solutions of the equation L d,1 w=0 and the eigenvalues of the linearized operators of the equations Δu+1/e2u(1−|u|2)=0, in B(0,1), about the radial solutions ud,e(x)=fd(r/e)eidθ, that tend to 0 as e tends to 0.

An a priori estimate for the singly periodic solutions of a semilinear equation

We consider the positive solutions u of -Delta u + u - u(p) = 0 in [ 0,2 pi] x RN - 1, which are 2 pi-periodic in x(1) and tend uniformly to 0 in the other variables. There exists a constant C such that any solution u verifies u( x(1), x(1)) <= Cw(0)(x(1)) where w(0) is the ground state solution of -Delta v + v - v(p) = 0 in RN - 1. We prove that exactly the same estimate is true when the period is 2 pi/epsilon, even when epsilon tends to 0. We have a similar result for the gradient.

The logarithmic Sobolev constant of the lamplighter

We give estimates on the logarithmic Sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base. Also, we give examples of application.

Solutions on a torus for a semilinear equation

We are interested in the positive doubly periodic solutions, which are even in each variable, of a stationary nonlinear Schrodinger equation in ℝ 2 , with a small parameter. For any pair of periods (2 a , 2 b ), we construct a branch of solutions that concentrate uniformly to the ground-state solution of the equation.

Remarks on solutions of a fourth-order problem

Abstract In this paper, we study the two following minimization problems: S 0 ( q , φ ) = inf u ∈ H 0 2 ( Ω ) , ‖ u + φ ‖ q = 1 ∫ Ω | Δ u | 2 and S θ ( q , φ ) = inf u ∈ H θ 2 ( Ω ) , ‖ u + φ ‖ q = 1 ∫ Ω | Δ u | 2 . We prove that for a class of maps φ , we have S θ ( q , φ ) S 0 ( q , φ ) and for another class, we have S θ ( q , φ ) = S 0 ( q , φ ) .