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

1. 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.