Nash–Moser theorem

The Nash–Moser theorem, attributed to mathematicians John Forbes Nash and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to a class of "tame" Fréchet spaces.

Introduction

In contrast to the Banach space case, in which the invertibility of the derivative at a point is sufficient for a map to be locally invertible, the Nash–Moser theorem requires the derivative to be invertible in a neighborhood. The theorem is widely used to prove local existence for non-linear partial differential equations in spaces of smooth functions. It is particularly useful when the inverse to the derivative "loses" derivatives, and therefore the Banach space implicit function theorem cannot be used.

History

While Nash (1956) originated the theorem as a step in his proof of the Nash embedding theorem, Moser (1966a, 1966b) showed that Nash's methods could be successfully applied to solve problems on periodic orbits in celestial mechanics.

Formal statement

The formal statement of the theorem is as follows:^[1]

Let

F

and

G

be tame Frechet spaces and let

P:U\subseteq F\rightarrow G

be a smooth tame map. Suppose that the equation for the derivative

DP(f)h=k

has a unique solution

h=VP(f)k

for all

f\in U

and all

k

, and that the family of inverses

VP:U\times G\rightarrow F

is a smooth tame map. Then P is locally invertible, and each local inverse

P^{{-1}}

is a smooth tame map.

References

↑ Hamilton, Richard S. (1982). "The inverse function theorem of Nash and Moser" (PDF-12MB). Bulletin of the American Mathematical Society. 7 (1): 65–222. doi:10.1090/S0273-0979-1982-15004-2. MR 0656198. . (A detailed exposition of the Nash–Moser theorem and its mathematical background.)

Moser, Jürgen (1966a), "A rapidly convergent iteration method and non-linear partial differential equations. I", Ann. Scuola Norm. Sup. Pisa (3), 20: 265–315, MR 0199523
Moser, Jürgen (1966b), "A rapidly convergent iteration method and non-linear partial differential equations. II", Ann. Scuola Norm. Sup. Pisa (3), 20: 499–535, MR 0206461
Nash, John (1956), "The imbedding problem for Riemannian manifolds", Annals of Mathematics, 63 (1): 20–63, doi:10.2307/1969989, JSTOR 1969989, MR 0075639 .

Functional analysis

Set / subset types	Absolutely convex Absorbing Balanced Bounded Convex Radial Star-shaped Symmetric Linear cone (subset) Convex cone (subset)

TVS types	Banach Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert LF-space Locally convex Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Webbed Topological tensor product (of Hilbert spaces)

Mapping topologies	Dual Dual space Operator Ultraweak Weak (operator) Mackey Strong (polar operator) Ultrastrong Uniform convergence

Linear operators	Adjoint Bilinear (form) Bounded / Unbounded Closed Continuous / Discontinuous Compact Fredholm Hilbert–Schmidt Functionals (positive) Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary

Set operations	Algebraic interior (core) Interior Minkowski addition Polar

Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition

Theorems	Banach–Alaoglu Banach–Saks Banach–Mazur Bessel's inequality Cauchy–Schwarz inequality Closed range Closed graph Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Open mapping Parseval's identity Schauder fixed-point

Analysis	Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Vector measure Weakly measurable function

This article is issued from Wikipedia - version of the 11/5/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.