Question

find the dirichlet green function of laplace equation for the interior of a colander with radius a

Answer 1

Since you have not specified which type of cylinder, hence I am assuming it to be 2D cylinder infinitely long

(the above figure is taken from http://physics.usask.ca)

For boundary value problems in which z-dependence is suppressed, it is convenient to formulate a two dimensional Green's function. Two dimensional Green's function for a long cylinder is to be found from

In the cylindrical geometry, the 2D gauss Greens function is given by

and the Green's function satisfies

makes the cylinder surface an equipotential surface at a potential

The desired Green's function is