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\title 
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Inverse problem for a parabolic equation with 
space-periodic boundary conditions by Carleman estimate
\endtitle
\leftheadtext{ Jongsung CHOI }
\rightheadtext{Inverse Parabolic Problem}
\author 
Jongsung Choi 
\endauthor
\affil
Department of Mathematical Sciences \\
The University of Tokyo \\
3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan  \endaffil        

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%===================================================================
\define\pde#1#2{\frac{\partial#1}{\partial#2}}
\define\wt#1{\widetilde{#1}}
%
\head Abstract \endhead
In $\Omega=\prod_{j=1}^n ]0,L_j[ \ \subset \Bbb{R}^n$  with  $L_j>0, 1\leq j \leq n$, we consider
$$
\left\{
\aligned
&u_t(t,x) = \Delta u(t,x)-p(x)u(t,x), \quad 0<t<T, \ x\in\Omega, \\
&u(0,x)=\mu (x), \quad x\in \Omega, \\
&\left.u\right|_{\Gamma_j^0}=\left.u\right|_{\Gamma_{j}^1} \ \text{and} \
\left.\pde{u}{x_j}\right|_{\Gamma_j^0}=\left.\pde{u}{x_j}\right|_{\Gamma_{j}^1}, \quad 1\leq j \leq n. 
\endaligned
\right.                                                            \tag1.1 $$
We denote by $\Gamma_j^0$ and $\Gamma_{j}^1$ the following parts of the boundary $\partial\Omega$:
$$
\Gamma_j^0=\partial \Omega \cap \{ x_j=0\},\quad \Gamma_{j}^1=\partial\Omega\cap\{x_j=L_j\}, \qquad j=1,\cdots,n.                                                   \tag1.2 $$
We assume that 
if $n=1$, then $\omega$ is an arbitrarily fixed subdomain of $\Omega=]0,L_1[$, 
and  if $n \geq 2$, then $\omega$ is an arbitrarily fixed subdomain, containing boundary of $\Gamma_j^i \ (i=0,1, \ j=1, \cdots, n)$ of $\Omega$, 
$\theta \in (0,T)$ is fixed.

Our inverse problem is to determine $p=p(x)$ and $\mu =\mu(x)$  from the measurement $u(p, \mu)|_{(0,T)\times \omega}$ and $u(p, \mu)(\theta,x)$, $x \in \Omega$. In this paper, we establish a conditional stability estimate for this inverse problem. 

The main ingredient in this paper is a Carleman estimate  and an argument used by Bukhgeim and Klibanov [BK].
The paper [BK] is an early works establishing the uniqueness in inverse problems, provided that a suitable Carleman estimate is found.
Our Carleman estimate in the case of periodic boundary conditions is proved along the idea in Frusikov and Imanuvilov [FI].


\Refs
\tenpoint
\widestnumber\key{AAAA}

\ref \key BK \by A.~L.~Bukhgeim and M.~A.~Klibanov
\paper Global uniqueness of a class of multidimensional inverse problems
\vol 24
\jour Sov. Math. Dokl.
\yr 1981
\pages 244--247
\endref


\ref \key FI \by A.~V.~Fursikov and O.~Yu.~Imanuvilov
\book Controllability of Evolution Equations 
\publ Lecture Notes Series No. 34
\publaddr Seoul National University, Seoul, Korea 
\yr 1996 
\endref


\endRefs  
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