Recovery of a source term or a speed with one measurement and applications (Q2849028)

The paper's aim is to study the problem of recovery of the source term \(a(t,x)F(x)\) in the wave equation in an anisotropic medium: NEWLINE\[NEWLINE(\partial^2_t-c^2 \Delta)w= a(t,x)F(x)\text{ in }(0,T) \times \mathbb R^nNEWLINE\]NEWLINE NEWLINE\[NEWLINEw|_{t=0}=0;\;\partial_t w|_{t=0}=0,NEWLINE\]NEWLINE (where \(a(t,x)\) is known such that \(a(0,x) \neq 0\)) with a single measurement. Carleman estimates combined with geometric arguments are used to give sharp conditions for uniqueness. Let \(c=1\) outside some domain \(\Omega\) with a smooth strictly convex boundary. Consider the problem NEWLINE\[NEWLINE(\partial^2_t-c^2 \Delta)u= 0\text{ in }(0,T) \times \mathbb R^nNEWLINE\]NEWLINE NEWLINE\[NEWLINEu|_{t=0}=0;\;\partial_t u|_{t=0}=f.NEWLINE\]NEWLINE Here \(c=c(x)>0\) and \(T>0\) is fixed. The problem of recovery of the sound speed \(c(x)\), given \(f\), and restricted to \([0,T] \times \partial \Omega\), is also studied. This inverse problem is clearly nonlinear. Sharp conditions for stability are given as well. An application to thermoacoustic tomography is also presented.