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{\it On Stability for an Inverse Hyperbolic Problem} 

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{\bf KiHyun Yun}\\ 
 Michigan State University\\
 {\tt kyun@math.msu.edu} 
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\noindent {\bf Abstract}:  In this talk, we focus on the inverse problem of
determining the potential $q$ by the Neumann to Dirichlet map $\Lambda_q$ in
the wave equation $u_{tt} - \Delta u + qu = 0$ in  $\Omega\times (0,T)$ with
$u(x,0)=u_t (x,0)=0$. Here, we establish a H\"{o}lder type stability with
exponent $1 -\epsilon $ as follows: for any small $\epsilon >0$, there exists
$\beta_0 > 0$ such that  
$$ \| {q_1 - q_2}_{L^{\infty} (\Omega)} \leq C \| {\Lambda_{q_1}-
  \Lambda_{q_2} }_* ^{ 1 - \epsilon}$$  
when $\| {q_1 -q_2 }_{H^{\beta}(\mathbb {R}^{n}) }\leq M$ for some $\beta
  >\beta_0$.  Here, $\| {\cdot}_* $ represents the operator norm.

To mention the previous related work, twenty years ago (1988), using Sylvester
and Uhlmann s method that is well known in inverse problems, Ziqi Sun, and
Alessandrini and Sylvester obtained $1/3-\epsilon$ Holder stability 
$$ \| {q_1 - q_2}_{L^{\infty} (\Omega)} \leq C \| {\Lambda_{q_1}-
  \Lambda_{q_2} }_* ^{ 1/3 - \epsilon}.$$ 
There has been no remarkable improvement in this result, even though Yamamoto
  made some results under the strong condition. 
              
As it mentioned in the abstract, we have obtained a nearly Lipschitz type
stability estimate under the same condition as Sun.

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