analysis.tex

\chapter{Analysis}

\section{Bump functions}
\label{sec:bump_functions}

\begin{thm}
  \label{thm:bump_function}
  For any two real numbers $0 < r < R$ there exists a smooth
  function $\phi\colon \set R^n \to \set R$ with
  \begin{align*}
    \forall p \in \set R^n: \phi(p) & \ge 0, &
    \forall p \in \set R^n: \norm p \leq r \implies \phi(p) & = 1, &
    \forall p \in \set R^n: \norm p \geq R \implies \phi(p) & = 0.
  \end{align*}
\end{thm}

A function as in the theorem is called a \emph{bump function}.

\section{Hadamard's lemma}

A subset $U$ of $n$-dimensional Euclidean space $\set R^n$ is \emph{star-shaped with
respect to a point $p \in U$} if
\[
  \forall x \in U \forall 0 \leq t \leq 1 :
  (1 - t) \, p + t \, x \in U.
\]
Any neighborhood of a point $p \in \set R^n$ contains an open neighborhood
star-shaped with respect to the point $p$ as the standard $\epsilon$-neighborhoods
are star-shaped with respect to their center (in fact, with respect to any point
of their interior).

\begin{thm}
  \label{thm:hadamard}
  Let $\phi$ be a smooth function defined on an open subset $U$ of $n$-dimensional
  Euclidean space $\set R^n$ that is star-shaped with respect to a point $a \in U$.
  Then there exist smooth functions $g_1$, \dots, $g_n$ on $U$ such that
  \[
    \forall x \in U: \phi(x) = \phi(a) + \sum_{i = 1}^n (x_i - a_i) \, g_i(x).
  \]
\end{thm}

\section{Ordinary differential equations}

Let $p_0 \in \set R^n$ be a point in Euclidean space and
$G \in \mathfrak U^0(p_0, \set R^n)$ an open subset. Let $f\colon G \to \set R^n$
be a smooth function.

\begin{thm}[Existence theorem for smooth local flows]
  \label{thm:diff_flow}
  There exists a neighborhood $U \in \mathfrak U^0(p_0, G)$, an
  $\epsilon \in \set R_+$ and a smooth function $\Psi\colon
  U_\epsilon(0) \times U \to G$ such that the curves
  \[
    \Psi^p\colon U_\epsilon(0) \to G, t \mapsto \Psi(t, p)
  \]
  solve the initial value problem
  \begin{align*}
    (\Psi^p)' & = f \circ \Psi^p, &
    \Psi^p(0) & = 0
  \end{align*}
  for each $p \in U$.
\end{thm}