  We study regularity properties of solutions to the Dirichlet problem for the
complex Homogeneous Monge-Amp\`ere equation. We show that for certain boundary
data on $\mathbb P^1$ the solution $\Phi$ to this Dirichlet problem is
connected via a Legendre transform to an associated flow in the complex plane
called the Hele-Shaw flow. Using this we determine precisely the harmonic discs
associated to $\Phi$. We then give examples for which these discs are not dense
in the product, and also prove that this situation persists after small
perturbations of the boundary data.
