A long cylinder of radius $R$ is made from two different material. Its radius $r<r_0$ $(r_0<R)$ part is a material with superconducting transition temperature $T_1$, and its $r_0<r<R$ part is another material with superconducting transition temperature $T_2$ $(T_1<T_2)$.

Let us consider two different process.

First, if we low down the temperature to $T$ $(T_1<T<T_2)$, and then add magnetic field $\vec{H}$ parallel to the axis of the cylinder.

Second, if we add magnetic field which $\vec{H}$ parallel to the axis of the cylinder, and then low down the temperature to $T$ $(T_1<T<T_2)$.

In the first case, we can easily know that only the surface at $r=R$ has superconducting current due to Meissner effect.

In the second case, two surface both have superconducting current. The current on the surface $r=r_0$ appears to keep the magnetic flux unchanged in the area $r<r_0$.

Are the currents on the surface at $r=R$ the same? I have two different answers and I don't know which one is right.

One answer is that the currents are the same in two case. Because of Meissner effect, $\vec{B}=0$ in superconductivity, the currents are the same in two case, because only current on the surface at $r=R$ can affect the magnetic field in the superconductivity.

Another answer is different by considering the change of magnetic flux. the variations of flux are different in two case, in the first is $B \pi R^2$ and the second case is $B \pi (R^2-r_0^2)$, so the currents are different.

I am very confused.

That's an interesting question which I believe you're smart enough to resolve by yourself. It is just the problem of the London equation with a temperature dependent penetration length $\lambda_{1,2}\left(T_{1},T_{2}\right)$. You can find a lot of similar calculations in the book by London, Superfluids vol.1, macroscopic theory of superconductivity, 1961. Then it will be interesting to get your solution on these pages of course :-) –  FraSchelle Oct 16 '13 at 12:23