Nonconstant soluliuns of the partial differential equationwhereais analytic in the open unit disk and |a| <1, are orientation-preserving harmonic mappings. We consider the case whereais a finite Blaschke product, so that ‖a‖= 1, and describe the behavior at the boundary. Some applications include conditions under which the image is a polygon and a situation where a univalent solution does not exist that maps onto a prescribed domain with a normalization at three boundary points.