Partition function of 2D Coulomb gases with radially symmetric potentials and a hard wall

The large $N$ asymptotic expansion of the partition function for the normal matrix model is predicted to have special features inherited from its interpretation as a two-dimensional Coulomb gas. However for the latter, it is most natural to include a hard wall at the boundary of the droplet. We probe how this affects the asymptotic expansion in the solvable case that the potential is radially symmetric and the droplet is a disk or an annulus. We allow too for the hard wall to be strictly inside the boundary of the droplet. It is observed the term of order $\log N$, has then a different rational number prefactor to that when the hard wall is at the droplet boundary. Also found are certain universal (potential independent) numerical constants given by definite integrals, both at order $\sqrt{N}$, and in the constant term.