Rejeb Hadiji

Ferromagnetic thin multi-structures

Abstract In this paper, starting from the classical 3 D non-convex and nonlocal micromagnetic energy for ferromagnetic materials, we determine, via an asymptotic analysis, the free energy of a multi-structure consisting of a nano-wire in junction with a thin film and of a multi-structure consisting of two joined nano-wires. We assume that the volumes of the two parts composing each multi-structure vanish with the same rate. In the first case, we obtain a 1 D limit problem on the nano-wire and a 2 D limit problem on the thin film, and the two limit problems are uncoupled. In the second case, we obtain two 1 D limit problems coupled by a junction condition on the magnetization. In both cases, the limit problem remains non-convex, but now it becomes completely local.

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.

The effect of a discontinuous weight for a critical Sobolev problem

ABSTRACT We study the minimizing problem where is a smooth bounded domain of , and p a positive discontinuous function. We prove the existence of a minimizer under some assumptions.

A biharmonic problem with constraint involving critical Sobolev exponent

We are concerned with the following minimization problems, where Ω ⊂ R N , N > 4, is a smooth bounded domain, q c = 2 N /( N − 4), ϕ ∈ C (Ω) ∩ L q c (Ω) and . We show that, for ϕ ≢ 0, each infimum is achieved. Under suitable conditions on ϕ, we establish the following gap phenomenon, for q ≤ q c . Moreover, we study the limit behaviour of the minimizers, as q goes to q c , in the case ϕ ∈ H (Ω).

A nonlinear general Neumann problem involving two critical exponents

We discuss the existence of solutions of nonlinear problem involving,two critical Sobolev exponents. we will ll out the su cient conditions to nd solutions for the problem in presence of a nonlinear Neumann boundary data with a critical nonlinearity. \

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 ) .