Lipschitz stability in inverse parabolic problems by the Carleman estimate

We consider a system yt(t, x) = -Ay(t, x) + g(t, x) 0 < t < T, x epsilon Omega y(theta, x) = y(0)(x) x epsilon Omega with a suitable boundary condition, where Omega subset of R-n is a bounded domain, -A is a uniformly elliptic operator of the second order whose coefficients are suitably regular for (t, x), theta epsilon]0, T[ is fixed, and a function g(t, x) satisfies \gt(t, x)\ less than or equal to C\g(theta, x) \ for (t, x) epsilon [0, T] x <(Omega)over bar>. Our inverse problems are determinations of g using overdetermining data gamma(vertical bar]0,T[x omega) or {gamma(vertical bar]0, T[x Gamma 0), del gamma(vertical bar]0, T[x Gamma 0)}, where omega subset of Omega and Gamma(0) subset of partial derivative Omega. Our main result is the Lipschitz stability in these inverse problems. We also consider the determination of f = f(x), x epsilon Omega in the case of g(t, x) = f(x)R(t, x) with given R satisfying R(theta, .) > 0 on Omega. Finally, we discuss an upper estimation of our overdetermining data by means of f.