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\title 
\vskip -1.3cm
Inverse Hyperbolic Problem: uniqueness and stability
\endtitle
\leftheadtext{M. YAMAMOTO}
\rightheadtext{inverse hyperbolic problems}
\author Masahiro Yamamoto \endauthor
\affil  
        Department of Mathematical Sciences \\
        The University of Tokyo \\
        3-8-1 Komaba, Meguro, Tokyo 153 Japan \\
        e-mail : myama\@ms.u-tokyo.ac.jp   
\endaffil
\endtopmatter
\document
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%\subhead Abstract \endsubhead
%\flushpar
With this lecture, we will discuss inverse hyperbolic problems of determining
functions in the equations.  We presently take
the following initial/boundary value problem for a single hyperbolic 
equation of the second order:
$$
\rho(x)\ppp_t^2 u(x,t) = \Delta u(x,t) + p(x)\ppp_t u(x,t) - q(x)u(x,t)
+ f(x)R(x,t), \thinspace x \in \Omega, \thinspace 0 < t < T.           \tag1.1
$$
$$
\text{$u(x,0)$,  $\ppp_t u(x,0)$, $x \in \Omega$ : given}.        \tag1.2
$$
$$
\text{the Dirichlet case : $u_{\vert\ppp\Omega\times (0,T)}$ is given}. \tag1.3
$$
$$
\text{the Neumann case: $\frac{\ppp u}{\ppp\nu}_{\vert\ppp\Omega\times (0,T)}$
is given}.                                                           \tag1.4
$$
Here and henceforth, let $\Omega \subset \Bbb R^n$ ($n \ge 1$) be a bounded 
domain whose boundary $\partial\Omega$ be of class $C^2$.  We set
$\ppp_t = \frac{\ppp}{\ppp t}$, $\ppp_t^2 u = \frac{\ppp^2}{\ppp t^2}$,
$\ppp_i = \frac{\ppp}{\ppp x_i}$, $\Delta = \sum^n_{i=1}\ppp_i^2$,
$\nabla = (\ppp_1, ..., \ppp_n)$, $\nu = \nu(x)$ the outward normal vector to
$\ppp\Omega$ at $x$ and $\frac{\ppp}{\ppp\nu} = \nabla\cdot\nu$, $R$ is
given.
\par
In this report, we survey
\subhead Inverse hyperbolic problem with a single measurement\endsubhead
Let $D \subset \overline{\Omega}$ be suitably given and let $T > 0$.
Determine some (or one) of the functions $\rho$, $p$, $q$ and $f$ from
observations $u_{\vert D\times (0,T)}$ or $\frac{\ppp u}{\ppp\nu}_
{\vert D\times (0,T)}$.
\par
In this report, we will exclusively discuss formulation with a single
measurement.  For the mathematical point of view, we can point the following
two topics for the inverse problem: {\bf the uniqueness} and
{\bf the stability}.
\par
The main purpose of this report is to show recent results on the uniqueness and
the stability which have been obtained by the author and his colleagues.
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%14 December 2000
