Changing domains and a solidification problem with void
We consider Laplace's equation posed in a region with a hole. The exact location and shape of this hole is not specified a priori. Dirichlet data are specified on the exterior boundary of the region and a Robin boundary condition holds on the edge of the hole. The problem represents a mathematical model of an equilibrium thermal state following a phase change process in which a denser solid has frozen out of its less dense liquid with the formation of a void. Since the solid is more dense that the liquid, the combined volume of frozen solid and remaining liquid is less than that of the initial liquid (which is assumed to have filled a finite container) and a vapor bubble or void results. Energy balance considerations impose two auxiliary conditions to be satisfied. We establish the existence of admissible holes, i.e., voids, for which a solution to the problem exists and, in addition, the existence of an admissible void with minimal circumference.