Abstract.Consider the reaction-diffusion equation in R"xR+ : ut-h2Au + la dérivée d'un potentiel bistable à puits également profonds et h un petit paramètre. Pour une condition initiale possédant une interface, on donne un développement asymptotique d'ordre arbitrairement élevé, ainsi que des estimations d'erreur valides jusqu'à un temps en 0(h~2) . A l'ordre le plus bas, l'interface évolue normalement, à une vitesse proportionnelle à la courbure moyenne.

International audienceIn this paper, $\phi$ will denote a lower semicontinuous convex proper function from $\mathbb{R}^N = H$ to $\mathbb{R}\cup\{+\infty\}$. The effective domain of $\phi$ is the set dom $\phi=\{x\in\mathbb{R}^N | \phi(x)<+ \infty\}$. We shall suppose that the interior of dom $\phi$ in $\mathbb{R}^N$ is not empty, and $\phi\geq 0$. These two assumptions do not restrict the generality. The scalar product in $H$ is denoted by $(x,y)$

Abstract. We analyze the one-dimensional Ginzburg-Landau functional of superconductivity on a planar graph. In the Euler-Lagrange equations, the equation for the phase can be integrated, provided that the order parameter does not vanish at the vertices; in this case, the minimization of the GinzburgLandau functional is equivalent to the minimization of another functional, whose unknowns are a real-valued function on the graph and a finite set of integers.

We consider a mechanical system with impact and n degrees of freedom, written in generalized coordinates. The system is not necessarily Lagrangian. The representative point is subject to a constraint: it must stay inside a closed set K with boundary of class C 3 . We assume that, at impact, the tangential component of the impulsion is conserved, while its normal coordinate is reflected and multiplied by a given coefficient of restitution e ∈ [0, 1]: the mechanically relevant notion of orthogonality is defined in terms of the local metric for the impulsions (local cotangent metric). We define a numerical scheme which enables us to approximate the solutions of the Cauchy problem: this is an ad hoc scheme which does not require a systematic search for the times of impact. We prove the convergence of this numerical scheme to a solution, which yields also an existence result. Without any a priori estimates, the convergence and the existence are local; with some a priori estimates, the convergence and the existence are proved on intervals depending exclusively on these estimates. The technique of proof uses a localization of the scheme close to the boundary of K; this idea is classical for a differential system studied in the framework of flows of a vector field; it is much more difficult to implement here, because finite differences schemes are only approximately local: straightening the boundary creates quadratic terms which cause all the difficulties of the proof.