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Journal of the European Mathematical Society

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Volume 23, Issue 10, 2021, pp. 3323–3349
DOI: 10.4171/JEMS/1086

Published online: 2021-06-08

A surjection theorem for maps with singular perturbation and loss of derivatives

Ivar Ekeland[1] and Éric Séré[2]

(1) Université Paris-Dauphine, France
(2) Université Paris-Dauphine, France

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 $\varepsilon$ goes to zero. These equations are of the form $F_\varepsilon(u)=v$ with $F_\varepsilon(0)=0$, $v$ small and given, $u$ 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 $F$ and $v$ than earlier ones, such as those of Hörmander [17]. For singularly perturbed functionals $F_\varepsilon$, we allow $v$ 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.

Keywords: Inverse function theorem, loss of derivatives, singular perturbations, Nash–Moser theorem, Cauchy problem, nonlinear Schrödinger system, Ekeland’s variational principle

Ekeland Ivar, Séré Éric: A surjection theorem for maps with singular perturbation and loss of derivatives. J. Eur. Math. Soc. 23 (2021), 3323-3349. doi: 10.4171/JEMS/1086