A surjection theorem for maps with singular perturbation and loss of derivatives
Ivar Ekeland
Université Paris-Dauphine, FranceÉric Séré
Université Paris-Dauphine, France
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Abstract
In this paper we introduce a new algorithm for solving perturbed nonlinear functional equations which admit a right-invertible linearization, but with an inverse that loses derivatives and may blow up when the perturbation parameter goes to zero. These equations are of the form with , small and given, small and unknown. The main difference from the by now classical Nash–Moser algorithm is that, instead of using a regularized Newton scheme, we solve a sequence of Galerkin problems thanks to a topological argument. As a consequence, in our estimates there are no quadratic terms. For problems without perturbation parameter, our results require weaker regularity assumptions on and than earlier ones, such as those of Hörmander [17]. For singularly perturbed functionals , we allow to be larger than in previous works. To illustrate this, we apply our method to a nonlinear Schrödinger Cauchy problem with concentrated initial data studied by Texier–Zumbrun [26], and we show that our result improves significantly on theirs.
Cite this article
Ivar Ekeland, Éric Séré, A surjection theorem for maps with singular perturbation and loss of derivatives. J. Eur. Math. Soc. 23 (2021), no. 10, pp. 3323–3349
DOI 10.4171/JEMS/1086