### ${\mathcal{L}}^{2,\text{\Phi}}$ regularity for nonlinear elliptic systems of second order.

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We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ (${P}_{\lambda}$) ⎩ ${u}_{\mid \partial \Omega}=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem (${P}_{\lambda}$) admits a non-zero, non-negative strong solution ${u}_{\lambda}\in {\bigcap}_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0\u207a}\left|\right|{u}_{\lambda}{\left|\right|}_{{W}^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto {I}_{\lambda}\left({u}_{\lambda}\right)$ is negative and decreasing in ]0,λ*[, where ${I}_{\lambda}$ is the energy functional related to (${P}_{\lambda}$).

We study the noncompact solution sequences to the mean field equation for arbitrarily signed vortices and observe the quantization of the mass of concentration, using the rescaling argument.

In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for ${S}^{1}$-valued maps ${m}^{\text{'}}$ (the magnetization) of two variables ${x}^{\text{'}}$: ${E}_{\epsilon}\left({m}^{\text{'}}\right)=\epsilon \int {|{\nabla}^{\text{'}}\xb7{m}^{\text{'}}|}^{2}d{x}^{\text{'}}+\frac{1}{2}\int {\left||{\nabla}^{\text{'}}{|}^{-1/2}{\nabla}^{\text{'}}\xb7{m}^{\text{'}}\right|}^{2}d{x}^{\text{'}}$. We are interested in the behavior of minimizers as $\epsilon \to 0$. They are expected to be ${S}^{1}$-valued maps ${m}^{\text{'}}$ of vanishing distributional divergence ${\nabla}^{\text{'}}\xb7{m}^{\text{'}}=0$, so that appropriate boundary conditions enforce line discontinuities. For finite $\epsilon >0$, these line discontinuities are approximated by smooth transition layers, the so-called Néel walls. Néel...

The aim of this paper is to compare and realize three efficient iterative methods, which have mesh independent convergence, and to propose some improvements for them. We look for the numerical solution of a nonlinear model problem using FEM discretization with gradient and Newton type methods. Three numerical methods have been carried out, namely, the gradient, Newton and quasi-Newton methods. We have solved the model problem with these methods, we have investigated the differences between them...

We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms ${W}^{1,p}$ of the associated chemical potential fields are bounded uniformly, where $p\>\frac{n}{2}$ and $n$ is the dimension of the domain. We show that the limit interface as $\epsilon $ tends to zero is an integral varifold with a sharp integrability condition on the mean curvature.

This paper deals with the problem of finding positive solutions to the equation -∆[u] = g(x,u) on a bounded domain 'Omega' with Dirichlet boundary conditions. The function g can change sign and has asymptotically linear behaviour. The solutions are found using the Mountain Pass Theorem.