Determining coefficients in class of heat equations via boundary measurements

When Omega subset of R-N is a bounded domain, we consider the problem of id entifiability of the coefficients, rho, A, q in the equation rho (x)partial derivative (t)u - div(A(x)delu) + q (x) u = 0 from boundary measurements o n two pieces Gamma (in) and Gamma (out) of partial derivative Omega. Provid ed that Gamma (in) boolean AND Gamma (out) has a nonempty interior, and ass uming that f(t,sigma) is the given input datum for (t,sigma) is an element of (0, T) x Gamma (in) and that the corresponding output datum is the therm al flux A(sigma)delu(T-0,sigma) . n(sigma) measured at a given time T-0 for sigma is an element of Gamma (out), we prove that knowledge of all possibl e pairs of input-output data (f, A delu(T-o) . n\Gamma (out)) determines uniquely the boundary spectral data of the underlying elliptic o perator. Under suitable hypothesis on rho, A, q, their identifiability is t hen proved. The same results hold when a mean value of the thermal flux is measured over a small interval of time.