Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media
Let \(g\) be a Riemannian metric in a bounded domain \(\Omega\) in \(\mathbb{R}^3\) with smooth boundary, and \(\Delta_g\) be the associated Laplace-Beltrami operator. The hyperbolic Dirichlet-Neumann map \(\Lambda_g\) is the operator which maps a function \(f\in H^2_{\text{loc}}(\partial\Omega)\) such that \(f= 0\) for \(t<0\) to the Neumann data of the solution \(u\) of the boundary value problem \[ (\partial^2_t- \Delta_g)u= 0\quad\text{in }(0,\infty)\times \Omega,\quad u|_{t= 0}= \partial_t u|_{t= 0}= 0\quad\text{in }\Omega,\quad u|_{(0, \infty)\times \partial\Omega}= f. \] For a diffeomorphism \(\psi:\overline \Omega\to\overline\Omega\) with \(\psi|_{\partial\Omega}=\) identity, \(\psi^*g\) denotes the pullback of the metric \(g\) with respect to \(\psi\). The author proves that if two metrics \(g_1\) and \(g_2\) are sufficiently close to the Euclidean metric in the norm of \(C^{10,\mu}(\overline\Omega)\) with some \(\mu> 0\), then there exists a \(C^{11}\) diffeomorphism \(\psi\) as above such that \[ \|\psi^* g_1- g_2\|_{L^2(\Omega)}\leq C(\|\Lambda_{g_1}- \Lambda_{g_2}\|^\sigma_*+\|\Lambda_{g_1}- \Lambda_{g_2}\|^*_{**}) \] for any \(\sigma< 1/5\), where \(\|\cdot\|_*\) and \(\|\cdot\|_{**}\) are the norms as operators from \(H^1((0, T)\times \partial\Omega))\) to \(L^2((0,T)\times \partial\Omega)\) and \(e^{\sqrt{\varepsilon t}} H^2_0(\mathbb{R}_+\times \partial\Omega)\) to \(e^{\sqrt{\varepsilon t}} L^2(\mathbb{R}_+\times \partial\Omega)\) respectively.
scientific article; zbMATH DE number 1216764
scientific article