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Pré-Publication, Document De Travail Année : 2019

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

Résumé

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 $\epsilon$ goes to zero. These equations are of the form $F_\epsilon(u)=v$ with $F_\epsilon(0)=0$, v small and given, u small and unknown. The main difference with 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 Hormander. For singularly perturbed functionals, we allow v to be larger than in previous works. To illustrate this, we apply our method to a nonlinear Schrodinger Cauchy problem with concentrated initial data studied by Texier-Zumbrun, and we show that our result improves significantly on theirs.
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Dates et versions

hal-01924328 , version 1 (15-11-2018)
hal-01924328 , version 2 (18-01-2019)
hal-01924328 , version 3 (05-04-2020)
hal-01924328 , version 4 (11-04-2020)
hal-01924328 , version 5 (10-06-2021)

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Ivar Ekeland, Eric Séré. A surjection theorem for maps with singular perturbations and loss of derivatives. 2019. ⟨hal-01924328v2⟩
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