Properties of а pseudo-harmonic function on closed domain

Abstract

Let f be a pseudo-harmonic function defined on  $k-$connected oriented closed domain D whose boundary consists of closed Jordan curves. We remind that this class of functions coincides with continuous functions which have a finitely many critical points at the interior of D each of them is saddle point and finitely many local extrema on its boundary.

In this work, it is proved that closure of any component of family which is a diference between D and such connected components of level curves of critical or semiregular values of f which contain critical and boundary critical points is a closed domain having one of three types (a ring, a strip or a sector). For the first, its boundary consists of two connected components that have no common points with D and level curve at any inner point is homeomorphic to circle. As well as the second and third,  their boundaries have one connected component and their level curves at any inner point are homeomorphic to a closed segment. There is difference between a number of arcs of boundary curves. If a domain is a strip, then its boundary contains two arcs that belong either one or two different boundary curves. If a domain is a sector, then  its boundary contains one arc of some boundary curves. By author some statements used for main theorem proof  are proved.