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摘要
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 \(q = q_k, k = 1, 2\) .
- 出版日期2016
- 单位中国科学技术大学
