Stability of the solution to a two-dimensional inverse problem of electrodynamics (Q2455103)

The author considers the following second-order linear hyperbolic equation that is a special case of the 2D Maxwell equations: \[ u_{tt}+\sigma u_t- c^2(\Delta u+qu)= 2\delta(t) \delta(x\cdot\nu), \qquad u|_{t<0}=0. \] Here \(x\in\mathbb R^2\), \(\nu= (\nu_1,\nu_2)\) unit vector, such that \(u=u(x,t,\nu)\). Assuming that the support of \(c(x)-1\), \(\sigma(x)\), \(q(x)\) is contained in a circle \(B= \{x\in\mathbb R^2\), \(|x-x^0|<r\}\) and \(C(x)\), \(\sigma(x)\), \(q(x)\) are smooth in \(\mathbb R^2\), the author proves uniqueness and stability results for the identification of \(c(x)\), \(\sigma(x)\), \(q(x)\) from data \(\tau(x,\nu^{(k)})\), \(u(x,t,\nu^{(k)})\), \(\frac{\partial}{\partial n} u(x,t,\nu^{(k)})\) for three different values \(\nu^{(k)}\), \(k=1,2,3\). Here \(\tau(x,\nu)\) is a solution to the problem \[ |\nabla\tau|^2= c^{-2}(x), \qquad \tau|_{x\cdot\nu=0}=0. \]