On continuous dependence for an inverse initial boundary value problem for the wave equation
The author studies the continuous dependence of the coefficient q in an initial boundary value problem for the wave equation \(u_{tt}=\Delta u+qu\) on the Neumann to Dirichlet map for this problem. This is shown by first proving an orthogonality result for any two pairs \((q_ 1,u_ 1)\), \((q_ 2,u_ 2)\) of solutions. Then a result of Rakesh and Symes (1988) is used to construct a family of special solutions \(u_ 1\) and \(u_ 2\). Using these special solutions gives an estimate for the Fourier transform of \(q_ 1-q_ 2\). Under the assumption that \(q_ 1-q_ 2\in H^{\alpha}({\mathbb{R}}^ n)\) for some \(\alpha >0\) he finally derives the desired estimates.
scientific article; zbMATH DE number 4212656
scientific article