I am interested to know if there exists a Green's Function for the Heat Equation which depends on the cylindrical coordinates r and theta, and which satisfies Dirichlet Boundary Conditions (at which G=0) at the following boundaries:

r=a for theta between 0 and 2(pi), and

r=b>a for theta between 0 and an angle which is smaller than 2(pi).

I am looking for a Green's Function that exists not only between r=a and r=b, but which also exists for r>b as well.

Any information about the existence of such a Green's Function would be very much appreciated.

Thanks for your help.