Uniqueness and stability in multidimensional hyperbolic inverse problems
The author, first, considers the uniqueness and the stability for a multidimensional hyperbolic inverse problem, which is for determining \(y\) and \(f\) from the following system of equations \[ \partial^{2}_{t} y(x,t)=\Delta y -q(x)y+f(x)R(x,t), \quad x\in \Omega,\;0<t<T, \] \[ y(x,0)=\partial_{t}y(x,0)=0,\;x\in \Omega, \quad y|_{\partial \Omega}=0,\;t\in (0,T), \quad \frac{\partial y}{\partial \nu}\mid_{\partial \Omega}=z. \] Under the assumptions \(| R(x,0)| \geq r_0>0\) and \(R\in W^{3,\infty}(\Omega\times (0,T))\) the uniqueness is proved. Moreover, there exists a constant \(C>0\) such that \( C^{-1}\| f\| _{L^2(\Omega)}\leq \| \frac{\partial y(f)}{\partial\nu}\| _{H^1(0,T;L^2(\partial \Omega))}\leq C\| f\| _{L^2 (\Omega)}. \) The above results are used to deduce the uniqueness and the stability of determining the coefficient of the lowest-order derivative, as usual.
scientific article; zbMATH DE number 1272013
scientific article