Exact and numerical solutions of a free boundary problem with a reciprocal growth law

A two-dimensional free boundary problem is formulated in which the normal velocity of the boundary is proportional to the inverse of the gradient of a harmonic function $T$. The field $T$ is defined in a simply connected region which includes the point at infinity where it has a logarithmic singularity. The growth problem in which the boundary expands outwards is formulated both in terms of the Schwarz function of the boundary and a Polubarinova-Galin equation for the conformal map of the region from the exterior of the unit disk. An expanding free boundary is shown to be stable and explicit solutions for growing ellipses and a class of polynomial lemniscates are derived. Numerical solution of the Polubarinova-Galin equation is used to compute the evolution of the boundary having other initial shapes.