On an Area Problem in Conformal Mapping

We consider an area problem for quadrilaterals Q. The geometry of Q is partly prescribed and partly free; the module is to be fixed, and its area is to be minimized. If Q is sufficiently simple, we can solve the problem explicitly by using a modified Schwarz-Christoffel integral. The free boundary arc is represented by an integral. The corresponding area problem for doubly connected domains can be reduced to the area problem for quadrilaterals, if the given boundary component is a regular polygon. We give results of numerical experiments.