Oy. Imanuvilov et M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, INVERSE PR, 17(4), 2001, pp. 717-728
For the solution u (p) = u (p) (x, t) to partial derivative (2)(t)u, (x, t)
-Deltau (x, t) - p(x)u(x, t) = 0 in Omega x (0, T) and partial derivativeu
/partial derivativev\ partial derivative Omega x (0, T) = 0 with given u(.,
0) and partial derivative (t)u (., 0), we consider an inverse problem of d
etermining p(x), x epsilon Omega, from data u(\omegax(0,T)). Here Omega sub
set of R-n, n = 1, 2, 3, is a bounded domain, omega is a sub-domain of n an
d T > 0. For suitable co C n and T > 0, we prove an upper and lower estimat
e of Lipschitz type between \\P - q\\(L2(Omega)) and \\partial derivative (
t)(u(p) - u(q))\\(L2(omega (0,T))) + \\partial derivative (2)(t)(u(p) - u(q
))\\(L2(omegax(0,T))).