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Helsinki University of Technology \hfill Somersalo/Bingham \\
Mathematics

\section*{Mat-1.198 Scattering Theory}

\subsection*{3$^\textrm{rd}$ set of exercises, 12.2.2003}

\vfill

% \emph{%
% By doing the exercises, you can earn points that will be taken into account 
% when determining your grade for the course:
% Do these exercises in advance, mark them as solved
% on the list at the exercise session and be prepared to present
% your solution on the blackboard.
% }

\emph{N.B. There is no lecture on Monday, the 10$^\textrm{th}$ of February.}
\vfill

\begin{enumerate}

\vfill
\item
Let $\partial D$ be a bounded $C^2$ surface. Using the estimate
\[
  |n(y) \cdot (z - y)| \le L |z - y|^2
  \qquad
  \text{ for all } z, y \in \partial D,
\]
show that there is an $h_0 > 0$ such that in 
\[
  B = \big\{ x = z + h n(z) \,|\, z \in \partial D,\, |h| \le h_0 \big\}
\]
the \emph{boundary projection}
$
  z + h n(z) \mapsto z
$
is unique. Further, show that it is possible to choose $h_0$ so small that
\[
  |x_1 - x_2|^2 \ge \frac{1}{2} \, |z_1 - z_2|^2
  \qquad
  \text{ for all } x_1, x_2 \in B,
\]
where $z_j$ is the boundary projection of $x_j$.


\vfill
\item
Show that for $0 < \beta < 1$, we have the estimate 
\[
  \Big|\, t \ln \frac{1}{t} \Big| \le \frac{1}{1 - \beta} \, t^\beta
  \qquad
  \text{when } 0 < t < 1.
\]


\vfill
\item
\emph{(Jump relation, special case)}\;
Let
\[
  \Phi_0(x) = \frac{1}{4 \pi |x|}
\]
and define
\[
  w(x) 
  = \int_{\partial D} \frac{\partial \Phi_0}{\partial n(y)}(x - y) \,dS(y),
  \qquad
  x \not\in \partial D
\]
where $\partial D$ is a bounded $C^2$ surface. Prove that
\[
  w(x) = 
  \begin{cases}
    -1, \quad & x \in D \\
    0,       & x \in \R^3 \setminus \overline{D}.
  \end{cases}
\]

\vfill
\item
\emph{(Jump relation, continued)}\; 
Show that
\[
  w(x) = - \frac{1}{2}
  \qquad
  \text{when } x \in \partial D
\]
and $w$ is interpreted as an improper integral.


\end{enumerate}

\end{document}
