Let $W$ be a non-negative function of class ${\mathrm{C}}^3$ from ${ℝ}^2$ to $ℝ$, which vanishes exactly at two points $𝐚$ and $𝐛$. Let $S^1(𝐚,𝐛)$ be the set of functions of a real variable which tend to $𝐚$ at $-\infty$ and to $𝐛$ at $+\infty$ and whose one dimensional energy$$E_1\left(v\right)={\int }_{ℝ}\left[W\left(v\right)+|v^{\text{'}}{|}^2/2\right]\mathrm{d}x$$is finite. Assume that there exist two isolated minimizers $z_{+}$ and $z_{-}$ of the energy $E_1$ over $S^1(𝐚,𝐛)$. Under a mild coercivity condition on the potential $W$ and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at $z_{+}$ and $z_{-}$, it is possible to prove...

Let W be a non-negative function of class C3 from ${}^2$ to $$, which vanishes exactly at two points a and b. Let S1(a, b) be the set of functions of a real variable which tend to a at -∞ and to b at +∞ and whose one dimensional energy $$E_1\left(v\right)={\int }_{\left[W\left(v\right)+{\left|v^{\text{'}}\right|}^2/2\right]}x$$ is finite. Assume that there exist two isolated minimizers z+ and z- of the energy E1 over S1(a, b). Under a mild coercivity condition on the potential W and a generic spectral condition on the linearization of the one-dimensional Euler–Lagrange operator at z+ and...

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega$ is a smooth bounded domain of ${ℝ}^2$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_p)$, as $p\to +\infty$. Among other results we show, under some symmetry assumptions on $\Omega$, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty$, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville...

In this article, we derive a complete mathematical analysis of a coupled 1D-2D model for 2D wave propagation in media including thin slots. Our error estimates are illustrated by numerical results.

This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale, so that we can asymptotically reduce them to immersed polygonal fault interfaces and the model finally consists in a coupling between...

In questa nota dimostriamo stime asintotiche ottimali per le soluzioni deboli non negative del problema al contorno $$-{\mathrm{\Delta }}_{{\mathbb{H}}^n}u=u^{\left(Q+2\right)/\left(Q-2\right)}in\mathrm{\Omega },u=0\text{ in }\partial \mathrm{\Omega }.$$$-{\mathrm{\Delta }}_{{\mathbb{H}}^n}$


For a fixed bounded open set $\Omega \subset {ℝ}^N$, a sequence of open sets ${\Omega }_n\subset \Omega$ and a sequence of sets ${\Gamma }_n\subset \partial \Omega \cap \partial {\Omega }_n$, we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on ${\Omega }_n$, satisfying Neumann boundary conditions on ${\Gamma }_n$ and Dirichlet boundary conditions on $\partial {\Omega }_n\setminus {\Gamma }_n$. We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on ${\Omega }_n$ and ${\Gamma }_n$ locally.