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Rejeb Hadiji

Publications and source records attributed to Rejeb Hadiji.

14 recordsLinked to original sources
arXiv2026

Analytical Study of Minimizers of an N-Field System with Sphere-Valued Constraints

This paper investigates a variational model that describes a system of N coupled sphere-valued fields through a penalization term imposing the constraint that their sum coincides with a prescribed sphere value map. We analyze the asymptotic behavior and regularity of minimizers as the penalization parameter tends to infinity. A fundamental dichotomy is established according to the number of interacting fields. For N = 2, topological obstructions may prevent the existence of exact sphere-valued decompositions, leading to the blow-up of the minimal energy in the large-penalization regime. In contrast, for N $\ge$ 3, exact decompositions always exist, yielding uniform energy bounds and allowing the identification of the limiting constrained problem via $\Gamma$-convergence. We further derive an explicit characterization of the limiting energy by decomposing admissible configurations into an average field and fluctuation components. Finally, we establish a gap phenomenon showing that, under suitable topological assumptions, every minimizer is necessarily singular for sufficiently large values of the penalization parameter. These results highlight the fundamental role of topology in the asymptotic behavior and regularity of coupled sphere-valued variational systems and extend previous results on single-field models to the multi-field setting.

math.AP↗
arXiv2025

On the Convergence of Solutions for the Ginzburg-Landau Equation and System

Let $(u_\varepsilon)$ be a family of solutions of the Ginzburg--Landau equation with boundary condition $u_\varepsilon = g$ on $\partial \Omega$ and of degree $0$. Let $u_0$ denote the harmonic map satisfying $u_0 = g$ on $\partial \Omega$. We show that, if there exists a constant $C_1 > 0$ such that for $\varepsilon$ sufficiently small we have $\frac{1}{2} \int_\Omega |\nabla u_\ve|^2 dx \leq C_1 \leq \frac{1}{2} \int_\Omega |\nabla u_0|^2 dx,$ then $C_1 = \frac{1}{2} \int_\Omega |\nabla u_0|^2 dx$ and $u_\ve ~\to ~ u_0 \qin H^1(\Om)$. We also prove that if there is a constant $C_2$ such that for $\ve$ small enough we have $ \frac12 \int_\Om |\nabla u_\ve|^2 dx \geq C_2 > \frac12 \int_\Om |\nabla u_0|^2 dx,$ then $|u_{\ve}|$ does not converge uniformly to $1$ on $\overline{\Om} $. We obtain analogous results for both symmetric and non-symmetric two-component Ginzburg--Landau systems.

math.AP↗
arXiv2024

Existence results for problems involving non local operator with an asymmetric weight and with a critical nonlinearity

Recently, great attention has been focused on the study of fractional and non-local operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. Our problem is related to the fractional Yamabe problem. First, we study a non-local problem involving the fractional laplacian, a critical nonlinearity with a non-symmetric weight,and a pertubation __in a bounded domain. We show that in the case of a linear perturbation. Next, if the perturbation is nonlinear, we find non-ground-state solutions for the problem.

math.AP↗
arXiv2024

Quantization effects for multi-component Ginzburg-Landau vortices

In this paper, we are concerned with $n$-component Ginzburg-Landau equations on $\rtwo$.By introducing a diffusion constant for each component, we discuss that the $n$-component equations are different from $n$-copies of the single Ginzburg-Landau equations.Then, the results of Brezis-Merle-Riviere for the single Ginzburg-Landau equation can be nontrivially extended to the multi-component case.First, we show that if the solutions have their gradients in $L^2$ space, they are trivial solutions.Second, we prove that if the potential is square summable, then it has quantized integrals, i.e., there exists one-to-one correspondence between the possible values of the potential energy and $\nat^n$.Third, we show that different diffusion coefficients in the system are important to obtain nontrivial solutions of $n$-component equations.

math.AP↗
arXiv2023

A system with weights and with critical Sobolev exponent

In this paper, we investigate the minimization problem : $$ \inf_{ \displaystyle{\begin{array}{lll} u \in H_0^1(Ω), v \in H_0^1(Ω),\\ \quad \| u \|_{L^{q}} =1, \quad \| v \|_{L^{q}} = 1 \end{array}}} \left[ \frac{1}{2} \int_Ω a(x) \vert \nabla u(x) \vert^2dx + \displaystyle{ \frac{1}{2} \int_Ω b(x) \vert \nabla v (x)\vert^2dx } - λ\displaystyle{\int_Ω u(x)v (x)dx} \right] $$ where $q=\frac{2N}{N-2}$, $ N \geq 4$, $a$ and $b$ are two continuous positive weight functions. We show the existence of solutions of the previous minimizing problem under some conditions on $a$, $b$, the dimension of the space and the parameter $λ$.

math.AP↗
arXiv2022

On a system of multi-component Ginzburg-Landau vortices

We study the asymptotic behavior of solutions for $n$-component Ginzburg-Landau equations as $\ve \to 0$. We prove that the minimizers converges locally in any $C^k$-norm to a solution of a system of generalized harmonic map equations.

math.AP↗
arXiv2020

A Ginzburg-Landau type energy with weight and with convex potential near zero

In this paper, we study the asymptotic behaviour of minimizing solutions of a Ginzburg-Landau type functional with a positive weight and with convex potential near $0$ and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis-Merle-Rivière.

math.AP↗
arXiv2019

A nonlinear problem witha weight and a nonvanishing boundary datum

We consider the problem: $$\inf_{{u}\in {H}^{1}_{g}(Ω),\|u\|_{q}=1} \int_Ω{p(x)}|\nabla{u(x)}|^{2}dx-λ\int_Ω| u(x)|^{2}dx$$ where $Ω$ is a bounded domain in $\R^{n}$, ${n}\geq{4}$, $ p : \barΩ\longrightarrow \R$ is a given positive weight such that $p\in H^{1}(Ω)\cap C(\barΩ)$, $0< c_1 \leq p(x) \leq c_2$, $λ$ is a real constant and $q=\frac{2n}{n-2}$ and $g$ a given positive boundary data. The goal of this present paper is to show that minimizers do exist. We distinguish two cases, the first is solved by a convex argument while the second is not so straightforward and will be treated using the behavior of the weight near its minimum and the fact that the boundary datum is not zero.

math.AP↗
arXiv2017

Existence of Solutions of a Non-Linear Eigenvalue Problem with a Variable Weight

We study the non-linear minimization problem on $H^1_0(Ω)\subset L^q$ with $q=\frac{2n}{n-2}$, $α>0$ and $n\geq4$~: \[\inf_{\substack{u\in H^1_0(Ω) \|u\|_{L^q}=1}}\int_Ωa(x,u)|\nabla u|^2 - λ\int_Ω |u|^2.\] where $a(x,s)$ presents a global minimum $α$ at $(x_0,0)$ with $x_0\inΩ$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behaviour of $a(x,s)$ with respect to $s$. The model case is \[\inf_{\substack{u\in H^1_0(Ω) \|u\|_{L^q}=1}}\int_Ω(α+|x|^β|u|^k)|\nabla u|^2 - λ\int_Ω |u|^2.\] In a previous paper dedicated to the same problem with $λ=0$, we showed that minimizers exist only in the range $β kn/q + 2$, minimizers do exist.

math.AP↗
arXiv2017

The effects of a discontinues weight for a problem with a critical nonlinearity

We study the minimizing problem $\inf\left\{\displaystyle\int_Ωp(x)|\nabla u|^{2}dx,\,u\in H^{1}_{0}(Ω),\,\|u\|_{L^{\frac{2N}{N-2}}(Ω)}=1\right\}$ where $Ω$ is a smooth bounded domain of $\R^{N}$, $N\geq 3$ and $p$ a positive discontinuous function. We prove the existence of a minimizer under some assumptions.

math.AP↗
arXiv2014

Ferromagnetic thin multi-structures

In this paper, starting from the classical 3D 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 same rate. In the first case, we obtain a 1D limit problem on the nano-wire and a 2D limit problem on the thin film, and the two limit problems are uncoupled. In the second case, we obtain two 1D 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.

math-ph↗
arXiv2014

A nonlinear general Neumann problem involving two critical

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. \

math.AP↗
arXiv2011

Non-Linear Effects in a Yamabe-Type Problem with Quasi-Linear Weight

We study the quasi-linear minimization problem on $H^1_0(Ω)\subset L^q$ with $q=\frac{2n}{n-2}$~: $$\inf_{\|u\|_{L^q}=1}\int_Ω(1+|x|^β|u|^k)|\nabla u|^2.$$ We show that minimizers exist only in the range $β<kn/q$ which corresponds to a dominant non-linear term. On the contrary, the linear influence for $β\geq kn/q$ prevents their existence.

math.AP↗