A Stability Estimate for an Inverse Problem of Determining a Coefficient in a Hyperbolic Equation with a Point Source

Abstract

For the solution to $\partial ^2_tu(x,t)-\triangle u(x,t)+q(x)u(x,t)=\delta (x,t)$ and $u\mid _{t<0}=0$, consider an inverse problem of determining $q(x), x\in \Omega $ from data $f=u\mid _{S_T}$ and $g=(\partial u/\partial \mathbf {n})\mid _{S_T}$. Here $\Omega \subset \{(x_1,x_2,x_3)\in \mathbb {R}^3\mid x_1>0\}$ is a bounded domain, $S_{T}=\{(x,t)\mid x\in {\partial \Omega },\vert {x}\vert<t<T+\vert {x}\vert \}$, $\mathbf {n}=\mathbf {n}(x)$ is the outward unit normal $\mathbf {n}$ to $\partial \Omega $, and $T>0$. For suitable $T>0$, prove a Lipschitz stability estimation:

$\begin{aligned} \left\| {q_1-q_2}\right\| _{L^2(\Omega )}\le C\left\{ \left\| {f_1-f_2}\right\| _{H^1(S_T)}+\left\| {g_1-g_2} \right\| _{L^2(S_T)}\right\} , \end{aligned}$

provided that $q_1$ satisfies a priori uniform boundedness conditions and $q_2$ satisfies a priori uniform smallness conditions, where $u_k$ is the solution to problem (1.1) with $q = q_k, k = 1, 2$.

Keywords

Inverse problem / Stability / Carleman estimate / Hyperbolic equation

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Funding

University of Science and Technology of China(YZ3471500002)