报 告 人：Eric SERE 教授
工作单位：巴黎第九大学-巴黎文理联大(Paris IX Dauphine-Paris Sciences et Lettres)
This is joint work with Ivar Ekeland. The Nash-Moser theorem allows to solve a functional equation F(u)=0 in a "scale" of Banach spaces, assuming that F(0) is very small and that near 0 the differential DF has a right inverse losing derivatives. The classical proof uses a Newton iteration scheme, which converges when F is of class C^2. In contrast, we only assume that F is continuous and has a G?teau first differential, which is right-invertible with loss of derivatives. In our iteration scheme, each step consists in solving a Galerkin approximation of the equation, using Ekeland's variational principle. We apply our method to a singular perturbation problem with loss of derivatives studied by Texier-Zumbrun. We compare the two results and we show that ours improves significantly on theirs, when applied, in particular, to a nonlinear Schrodinger Cauchy problem with highly oscillatory initial data: we are able to deal with larger oscillations.