An Inverse Function Theorem in Frechet Spaces
By Ivar Ekeland. 2010
74 citations recorded by the source corpus.
I present an inverse function theorem for differentiable maps between Frechet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C^2, or even C^1, or even Frechet-differentiable.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/f1f1d9429872d886913e0c45f0c7f38ed0e21329
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Computational Theory and Mathematics
In Statistical and Nonlinear Physics
- Massimiliano Berti
- P. Bolle
- Nicolas Bonnotte
- Michela Procesi
- Riccardo Montalto
- Ivar Ekeland
- Pierre Baldi
- Livia Corsi
Citations in the corpus
- Sobolev Periodic Solutions of Nonlinear Wave Equations in Higher Spatial Dimensions (2010, methodology): “This is also related to a famous counter-example of Lojasiewicz-Zehnder [19] concerning the optimal conditions in an abstract Nash-Moser implicit function theorem.”
- Unidimensional and Evolution Methods for Optimal Transportation (2013, methodology): “1Ivar Ekeland recently showed this is not the only way, by proving a more general inverse function theorem using his variational principle instead of Newton’s method [26].”
- Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential (2010, methodology): “14) are optimal for the convergence of the Nash-Moser iterative scheme, as a famous counter-example of Lojasiewicz-Zehnder [32] shows: if δ = 1 the scheme does not converge.”