Abstract:
We numerically study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions having the property of being exactly translated by $p$ sites along the chain after a fixed propagation time $T$ (these solutions generalize the concept of solitary waves for which $p=1$). In the case of even on-site potentials, the existence of small amplitude travelling breathers superposed on a small oscillatory tail has been proved recently (G. James and Y. Sire, to appear in {\sl Comm. Math. Phys.}, 2004), the tail being exponentially small with respect to the central oscillation size. In this paper we compute these solutions numerically and continue them into the large amplitude regime for different types of even potentials. We find that Klein-Gordon chains can support highly localized travelling breather solutions superposed on an oscillatory tail. We provide examples where the tail can be made very small and is difficult to detect at the scale of central oscillations. In addition we numerically observe the existence of these solutions in the case of non even potentials.

Abstract:
We deal with symmetry properties for solutions of nonlocal equations of the type $(-\Delta)^s v= f(v)\qquad {in $\R^n$,}$ where $s \in (0,1)$ and the operator $(-\Delta)^s$ is the so-called fractional Laplacian. The study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation ${-div (x^\a \nabla u)=0 \qquad {on $\R^n\times(0,+\infty)$} -x^\a u_x = f(u) \qquad {on $\R^n\times\{0\}$} $ where $\a \in (-1,1)$. This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator $\Gamma_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $ is $(-\Delta)^{\frac{1-\a}{2}}$. This equation is related to the fractional Laplacian since the Dirichlet-to-Neumann operator $\Gamma_\a: u|_{\partial \R^{n+1}_+} \mapsto -x^\a u_x |_{\partial \R^{n+1}_+} $ is $(-\Delta)^{\frac{1-\a}{2}}$. More generally, we study the so-called boundary reaction equations given by ${-div (\mu(x) \nabla u)+g(x,u)=0 {on $\R^n\times(0,+\infty)$} - \mu(x) u_x = f(u) {on $\R^n\times{0}$}$ under some natural assumptions on the diffusion coefficient $\mu$ and on the nonlinearities $f$ and $g$. We prove a geometric formula of Poincar\'e-type for stable solutions, from which we derive a symmetry result in the spirit of a conjecture of De Giorgi.

Abstract:
The goal of this paper is to investigate some rigidity properties of stable solutions of elliptic equations set on manifolds with boundary. We provide several types of results, according to the dimension of the manifold and the sign of its Ricci curvature.

Abstract:
This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some conical singularities and maximizes the first eigenvalue in the conformal class of the background metric. We also prove that the map associating a finite number of eigenvectors of the maximizing $\lambda_1$ into the sphere is harmonic, establishing a link between conformal spectrum and harmonic maps.

Abstract:
We prove an algebra property under pointwise multiplication for Besov spaces defined on Lie groups of polynomial growth. When the setting is restricted to the case of H-type groups, this algebra property is generalized to paraproduct estimates.

Abstract:
We consider a functional obtained by adding a trace term to the Allen-Cahn phase segregation model and we prove some density estimates for the level sets of the interfaces. We treat in a unified way also the cases of possible degeneracy and singularity of the ellipticity of the model and the quasiminimal case.

Abstract:
This is the first of two articles dealing with the equation $(-\Delta)^{s} v= f(v)$ in $\mathbb{R}^{n}$, with $s\in (0,1)$, where $(-\Delta)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a L\'evy process. This equation can be realized as a local linear degenerate elliptic equation in $\mathbb{R}^{n+1}_+$ together with a nonlinear Neumann boundary condition on $\partial \mathbb{R}^{n+1}_+=\mathbb{R}^{n}$. In this first article, we establish necessary conditions on the nonlinearity $f$ to admit certain type of solutions, with special interest in bounded increasing solutions in all of $\mathbb{R}$. These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian ---in the spirit of a result of Modica for the Laplacian. In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.

Abstract:
We study the $\Gamma$-convergence of the following functional ($p>2$) $$ F_{\epsilon}(u):=\epsilon^{p-2}\int_{\Omega}|Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\epsilon^{\frac{p-2}{p-1}}}\int_{\Omega}W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\epsilon}}\int_{\partial\Omega}V(Tu)d\mathcal{H}^2, $$ where $\Omega$ is an open bounded set of $\mathbb{R}^3$ and $W$ and $V$ are two non-negative continuous functions vanishing at $\alpha, \beta$ and $\alpha', \beta'$, respectively. In the previous functional, we fix $a=2-p$ and $u$ is a scalar density function, $Tu$ denotes its trace on $\partial\Omega$, $d(x,\partial \Omega)$ stands for the distance function to the boundary $\partial\Om$. We show that the singular limit of the energies $F_{\epsilon}$ leads to a coupled problem of bulk and surface phase transitions.

Abstract:
This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -\Delta u = f & \text{in} \Omega u= 0 & \text{on} \partial \Omega$$ where $f$ belongs to some $L^p(\Omega)$ and $\Omega$ is a Reifenberg-flat domain of $\mathbb R^n.$ More precisely, we prove that given an exponent $\alpha\in (0,1)$, there exists an $\varepsilon>0$ such that the solution $u$ to the previous system is locally H\"older continuous provided that $\Omega$ is $(\varepsilon,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem.

Abstract:
We study the time of existence of the solutions of the following Schr\"odinger equation $$i\psi_t = (-\Delta)^s \psi +f(|\psi|^2)\psi, x \in \mathbb S^d, or x\in\T^d$$ where $(-\Delta)^s$ stands for the spectrally defined fractional Laplacian with $s>1/2$ and $f$ a smooth function. We prove an almost global existence result for almost all $s>1/2$.