Dirichlet BC:

T(r=R1,z)=T1, T(r=R2,z)=T2, T(r,Z1)=T3, T(r,Z2)=T4.

Newman BC:

In this case, because you look for steady state solution, all the heat that enters the system must leave it, therefore you can only have three BC's, such as for example:

-k*dT/dr(r=R1,z)=Q1

-k*dT/dz(r,z=Z1)=Q2

-k*dT/dz(r,z=Z2)=Q3

Then you will have an extra degree of freedom, which you can set by applying normalization condition such as that the integral of the energy over the entire volume is constant.

I agree, this is a classical elliptic equation that requires to prescribe Dirichlet, Neumann or Robin (linear combination of the previous) BC.s along all the boundaries to have a mathematically well posed Laplace problem.

Thank you Yaron Ben Ami for your answer