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\begin{document}
\title{Reconstruction of Conductivity and its Derivatives at the Boundary 
from the Localized Dirichlet to Neumann Map}
\author{Gen NAKAMURA\thanks{Department of Mathematics, Faculty of 
Engineering, Gunma University,Japan.} \
and Kazumi TANUMA\thanks{Division of Mathematical Sciences, Osaka Kyoiku 
University, Japan.}}
\date{}
\maketitle

Let $\Omega\in R^n\ (n\ge 2)$ be a bounded domain with Lipschitz boundary 
$\partial\Omega$.
Physically $\Omega$ is considered as an isotropic, static and conductive 
medium with conductivity $\gamma\in L^\infty(\Omega)$.
When an electric potential $f\in H^{1/2}(\partial\Omega)$ is applied to the 
boundary $\partial\Omega$,
the potential $u$ solves the Dirichlet problem
\begin{equation}
\nabla\cdot(\gamma\nabla u)=0\quad\mbox{in}\quad\Omega,
\qquad
u|_{\partial\Omega}=f.
\end{equation}

Assume that there is a constant $\delta>0$ such that 
$\gamma(x)\ge\delta\quad(\mbox{a.e.}\ x\in\Omega)$.
Then, there exists a unique weak solution $u\in H^1(\Omega)$ to (1).
Define the Dirichlet to Neumann map $\Lambda_\gamma\ :\ 
H^{1/2}(\partial\Omega) \longrightarrow H^{-1/2}(\partial\Omega)$ by
\begin{equation}
<\Lambda_\gamma f,\ g>=\int_\Omega \gamma\nabla u\cdot\nabla v\ dx 
\quad(g\in H^{1/2}(\partial\Omega)),
\end{equation}
where $u$ is the solution to (1), $v$ is any $v\in H^1(\Omega)$ satisfying 
$v\vert_{\partial\Omega}=g$ and $<\ ,\ >$ is
the bilinear pairing between $H^{1/2}(\partial\Omega)$ and 
$H^{-1/2}(\partial\Omega)$.
Note that $\Lambda_\gamma f=\gamma\nabla u\cdot n$ when $f\in 
H^{3/2}(\partial\Omega)$, $\gamma\in C^1(\overline\Omega)$ and
$\partial\Omega$ is $C^2$, where $n$ is the unit outer normal to 
$\partial\Omega$.
Hence  $\Lambda_\gamma f$ is the current flux across $\partial\Omega$ 
produced by the potential $f$ on $\partial\Omega$.
\par
The problem of determining conductivity of the medium from the measurements of
the electric potential on the boundary and the corresponding current flux 
across the boundary is expressed as
{\bf Inverse Problem} ``Determine $\gamma(x)$ from $\Lambda_\gamma$''.
Since this problem was posed by
  A.P.Calderon, many results on uniqueness, stability, reconstruction have been
proved.

In this talk, letting $x_0$ be any point on $\partial\Omega$,
  we give a formula for reconstructing $\gamma$ and its normal derivative at 
$x_0\in\partial\Omega$
{\it simultaneously}
from the localized $\Lambda_\gamma$ around $x_0$.
In our new formula we need  only some regularity assumptions on $\gamma$ and on
$\partial\Omega$ around $x_0$, and we take as the Dirichlet data the 
highly oscillating functions which are compactly supported in a
small neighborhood of $x_0$ on $\partial\Omega$.


\end{document}
