On Stability for an Inverse Hyperbolic Problem

KiHyun Yun
Michigan State University

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 Hölder type stability with exponent $ 1 -\epsilon $ as follows: for any small $ \epsilon >0$, there exists $ \beta_0 > 0$ such that

$\displaystyle \Vert {q_1 - q_2}_{L^{\infty} (\Omega)} \leq C \Vert {\Lambda_{q_1}-
\Lambda_{q_2} }_* ^{ 1 - \epsilon}$

when $ \Vert {q_1 -q_2 }_{H^{\beta}(\mathbb{R}^{n}) }\leq M$ for some $ \beta
>\beta_0$. Here, $ \Vert {\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

$\displaystyle \Vert {q_1 - q_2}_{L^{\infty} (\Omega)} \leq C \Vert {\Lambda_{q_1}-
\Lambda_{q_2} }_* ^{ 1/3 - \epsilon}.$

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.