On the location of critical point for the Poisson equation in plane

The location of the unique critical point of Δu=−1 is investigated by conformal mapping method in complex variables. It is found that if the domain is given by r=1+ϵp(θ), the critical point coincides with the center of mass up to the order of ϵ. However, the two do not exactly match in general as shown by simple examples.