Asymptotic Analysis

Cet article est consacré aux systèmes de deux lois de conservation. Les conditions initiales sont choisies sous la forme d'oscillations de faible amplitude. On prouve que les solutions BV obtenues à l'aide du schéma de Glimm convergent vers l'unique solution entropique des équations de modulation de l'optique géométrique non linéaire.

Using the formal approach of asymptotic expansions we identify the two-dimensional limit membrane and bending models in the theory of linearly elastic shells. In this fashion, we retrieve the two asymptotic behaviors already announced in a short communication based on a multiple-scale technique.

In this paper we study the local behavior near zero for the solutions of a class of second order differential equations which include, as a particular case, the radial version of the partial nonlinear differential equation, involving the m-Laplacian operator,

div(|∇u|^m−2∇u)+f(|x|,u)=0, x∈Rⁿ\{0},

where n>m>1. Some important examples of f are given by

f(t,u)=Ct^νu|u|^q−2  and  f(u)=C₁u|u|^p−2+C₂u|u|^q−2.

As an application of our general results to the first case we extend to m>1 and ν>−m some estimates given in the literature for the Laplacian case m=2 or for ν=0.