An Inverse Problem for the Heat Equation II

Completeness of the set of products of the derivatives of the solutions to the equation (av′)′ − v = 0, v(0, ) = 0 is proved. This property is used to prove the uniqueness of the solution to an inverse problem of finding conductivity in the heat equation $ \dot u = (a(x)u')' $, u(x, 0) = 0, u(0, t) = 0, u(1, t) = f(t) known for all t > 0, from the heat flux a(1)u′(1, t) = g(t). Uniqueness of the solution to this problem is proved. The proof is based on Property C. It is proved the inverse that the inverse problem with the extra data (the flux) measured at the point, where the temperature is kept at zero, (point x = 0 in our case) does not have a unique solution, in general.