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 epsil on Omega y(theta, x) = y(0)(x) x epsilon Omega with a suitable boundar y condition, where Omega subset of R-n is a bounded domain, -A is a un iformly elliptic operator of the second order whose coefficients are s uitably regular for (t, x), theta epsilon]0, T[ is fixed, and a functi on 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 ar e 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 stabilit y 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 e stimation of our overdetermining data by means of f.