On a Problem of the Constructive Theory of Harmonic Mappings

Abstract

The problem of irremovable error appears in finite difference realization of the Winslow approach in the constructive theory of harmonic mappings. As an example, we consider the well-known Roache-Steinberg problem and demonstrate a new approach, which allows us to construct harmonic mappings of complicated domains effectively and with high precision. This possibility is given by the analytic-numerical method of multipoles with exponential convergence rate. It guarantees effective construction of a harmonic mapping with precision controlled by an a posteriori estimate in a uniform norm with respect to the domain.

Singular behavior of harmonic maps near corners

“…However, it is not guaranteed that the map constructed that way is one-sheeted; this is easily confirmed by the corresponding counterexamples (see, e. g., [8]). Note that, for conformal maps (unlike harmonic ones), the homeomorphism of the boundaries of Jordan domains implies the homeomorphism of the (closed) domains themselves (see the Caratheodory theorem in [9]).…”

“…For the map F(z) defined by (2), a sufficient condition to be one-sheeted is the convexity of the domain Z (see the Radó-Kneser-Choquet theorem in [3], [5]- [7]). Harmonic maps are broadly investigated (see [1]- [4], [8], [10]- [12]). The interest to this direction is caused by its great theoretical importance.…”