turkmath.org

Harmonic mappings dened on planar domains are the sum of $U(x, y)+iV (x, y)$ where $U$ and $V$ are harmonic functions. We study problems related to sense-preserving univalent harmonic mappings. Such mappings are locally quasiconformal and their second dilatation is an analytic function bounded by one. If the dilataion of such a function is a square of an analytic function, then they naturally produce a nonparametric minimal surface. We shall discuss two lately solved open problems, one on simply connected domain and the other in doubly connected domain. The problems were posed by Sheil-Small and Ivaniec, Kovalev, and Onninen.