Dans ce papier on étudie l’existence et le comportement asymptotique des solutions de type ondes progressives à propagations finies de l’équation . On prouve que ces solutions existent si et seulement si et ou bien et . On donne aussi le comportement asymptotique de ces solutions.
A class of (1 + 1)-dimensional nonlinear boundary value problems (BVPs), modeling the process of melting and evaporation of solid materials, is studied by means of the classical Lie symmetry method. A new definition of invariance in Lie's sense for BVP is presented and applied to the class of BVPs in question.
We consider the operator in with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound for any and . This includes a Lieb-Thirring inequality in the critical endpoint case.
Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one dimension. This allows us to improve on the known estimates of best constants in Lieb–Thirring inequalities for the sum of the negative eigenvalues for multidimensional Schrödinger operators.
In this talk, we describe some recent results on the Lifshitz behavior of the density of states for non monotonous random models. Non monotonous means that the random operator is not a monotonous function of the random variables. The models we consider will mainly be of alloy type but in some cases we also can apply our methods to random displacement models.