02-276 John Andrew Vano
A Nash-Moser Implicit Function Theorem with Whitney Regularity and Applications (850K, PDF) Jun 20, 02
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Abstract. This dissertation establishes the Whitney regularity with respect to parameters of implicit functions obtained from a Nash-Moser implicit function theorem. As an application of this result, we study the problem of wave propagation in resonating cavities. Using a modification of the general setup in [Zehnder75], we consider functionals $\F : U \times V \to Z$ which have an approximate right inverse $R : \calC \times V \to L(Z, Y)$. Here $U \subseteq X$ and $V \subseteq Y$ are open sets of scales of Banach spaces (scale parameters are suppressed here for brevity) and $\calC \subseteq U$ is an arbitrary set of parameters (in applications $\calC$ is often a Cantor set). Under appropriate hypothesis on $\F$, which are natural extensions of [Zehnder75], we show that given $(x_0,y_0)$ with $\F(x_0,y_0)=0$ for $x \in \calC$ near $x_0$ there exists a function $g(x)$, Whitney regular with respect to $x$, which satisfies $\F(x,g(x))=0$. The problem of wave propagation in a cavity with (quasi-periodically) moving boundary can be reduced to the study of a family of torus maps. Because of their extremely degenerate nature, this family is not covered by known versions of KAM theory. However, our implicit function theorem approach allows us to overcome these problems and prove a degenerate KAM theory. Our approach can also be applied to other problems of current interest.

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