### $\mathrm{\mathcal{B}\mathcal{C}}$-total stability and almost periodicity for some partial functional differential equations with infinite delay.

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The paper deals with the dimensional reduction from 2D to 1D in magnetoelastic interactions. We adopt a simplified, but nontrivial model described by the Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We identify the limit problem by using the so-called energy method.

In this paper we study the existence of critical points for noncoercive functionals, whose principal part has a degenerate coerciveness. A bifurcation result at zero for the associated differential operator is established.

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 questa Nota si dimostra un risultato enunciato nel § 5 della pubblicazione [4]. Per le soluzioni di un sistema ellittico base, con non-linearità $q\ge 2$, vale un principio di massimo analogo a quello dimostrato in [3] nel caso di non-linearità $q=2$.

In this paper a mixed boundary value problem for the fourth order hyperbolic equation with constant coefficients which is connected with response of semi-space to a short laser pulse» and belongs to generalized Thermoelasticity is studied. This problem was considered by R. B. Hetnarski and J. Ignaczak, who established some important physical consequences. The present paper contains proof of the existence, uniqueness and continuous dependence of a solution on the datum, together with an effective...

This paper is a set of lecture notes for a short introductory course on homogenization. It covers the basic tools of periodic homogenization (two-scale asymptotic expansions, the oscillating test function method and two-scale convergence) and briefly describes the main results of the more general theory of G− or H−convergence. Several applications of the method are given: derivation of Darcy’s law for flows in porous media, derivation of the porosity...