Global Uniqueness and Stability in Determining Coefficients of Wave Equations

We consider an inverse problem of determining p(x), x ∈ Ω in (∂2 u/∂t 2)(x, t) − Δu(x, t) − p(x)u(x, t) = 0 in Ω × (0, T) and (∂u/∂ν)|∂Ω×(0, T) = 0 with given u(·, 0) and (∂u/∂t)(·, 0). Here Ω ⊂ R n , n = 1, 2, 3, is a bounded domain. We prove the Lipschitz stability in determining p from u|∂Ω×(0,T), under the assumption that T is greater than the diameter of Ω and u(·,0) > 0 on . Our stability holds over Ω without any knowledge of boundary values of p. The proof is based on the Carleman estimate.