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

For the solution u(p) = u(p)(t, x) to ∂_t² 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|_{L² (Ω)} ≤ C( ∑ _{j = 2}³ |∂_t^j (u(p) − u(q))|_{L² ((0,T)×ω)}) ^κwith 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.