Global Lipschitz stability in an inverse hyperbolic problem by interior observations 论文

摘要

For the solution u(p) = u(p)(x,t) to ∂t2u(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 ∂tu(·,0), we consider an inverse problem of determining p(x), x∊Ω, from data u|ω×(0,T). Here Ω⊂n, n = 1,2,3, is a bounded domain, ω is a sub-domain of Ω and T>0. For suitable ω⊂Ω and T>0, we prove an upper and lower estimate of Lipschitz type between ||p-q||L2(Ω) and ||∂t(u(p)-u(q))||L2(ω×(0,T)) + ||∂t2(u(p)-u(q))||L2(ω×(0,T)).