Applications of the duality between the Complex Monge-Amp\`ere Equation and the Hele-Shaw flow

We give two applications of the the duality between the complex Homogeneous Monge-Amp\`ere Equation (HMAE) and the Hele-Shaw flow. First, we prove existence of smooth boundary data for which the weak solution to the Dirichlet problem for the HMAE over $\mathbb P^1\times \overline{\mathbb D}$ is not twice differentiable at a given collection of points, and also examples that are not twice differentiable along a set of codimension one in $\mathbb{P}^1\times \partial \mathbb{D}$. Second, we produce explicit families of smooth geodesic rays in the space of K\"ahler metrics on $\mathbb P^1$ and on the unit disc $\mathbb D$ that are constructed from an exhausting family of increasing smoothly varying simply connected domains.