Determination of a coefficient in an acoustic equation with a single measurement

For the solution u ( p ) = u ( p )( t , x ) to ∂ t 2 u ( t , x ) − div ( p ( x )∇ u ( t , x )) = 0 in (0, T ) × Ω with given u | (0, T ) × ∂Ω , u (0, ·) and ∂ t u (0, ·), we consider an inverse problem concerning the determination of the coefficient p ( x ), x ∊ Ω from data u | (0, T ) × ω . Here Ω ⊂ n is a bounded domain, and ω is some subdomain of Ω and T > 0. For suitable ω ⊂ Ω and T > 0, we prove an estimate of the Hölder type: | p − q | L 2 (Ω) ≤ C ( ∑ j = 2 3 |∂ t j ( u ( p ) − u ( q ))| L 2 ((0, T ) × ω) ) κ with some κ ∊ (0, 1), provided that p , q satisfy a priori uniform boundedness conditions, compatible conditions and some positivity conditions. The keys are Carleman estimates for a hyperbolic operator in an H −1 -space.