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 $\\varepsilon$ 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.
Highlights
The basic idea of the inverse function theorem ( IFT) is that, if a map F is differentiable at a point u0 and the derivative DF (u0) is invertible, the map itself is invertible in some neighbourhood of u0
Bolle and Procesi [10] prove a new version of the Nash-Moser theorem by solving a sequence of Galerkin problems Π′nF = Π′nv, un ∈ En, where Πn and Π′n are projectors and En is the range of Πn
The purpose of this paper has been to introduce a new algorithm into the "hard" inverse function theorem, where both DF (u) and its right inverse L (u) lose derivatives, in order to improve its range of validity
Summary
The basic idea of the inverse function theorem ( IFT) is that, if a map F is differentiable at a point u0 and the derivative DF (u0) is invertible, the map itself is invertible in some neighbourhood of u0. Bolle and Procesi [10] prove a new version of the Nash-Moser theorem by solving a sequence of Galerkin problems Π′nF (un) = Π′nv, un ∈ En, where Πn and Π′n are projectors and En is the range of Πn They find the solution of each projected equation thanks to a Picard iteration: un. In contrast with [10], the Newton steps are completely absent in our new algorithm, they are replaced by the topological argument from [12] (Theorem 2), ensuring the solvability of each projected equation This allows us to work with functionals F that are only continuous and Gâteaux-differentiable, while the standard Nash-Moser scheme requires twice-differentiable functionals.
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