On an Unboundedness Property of Solutions of Elliptic Systems in the Plane - Grigori Giorgadze; Giorgi Makatsaria; Nino Manjavidze (Mathematics)
The issue of the invariance of the unboundedness property of the solutions of the Carleman–Bers–Vekua system (generalized analytic functions) with respect to the transformation of the restriction is studied. The concept of the rating of an unbounded continuous function is introduced. A continuous unbounded function of zero rating is constructed, whose restriction to every strip of the plane is bounded. For entire and generalized entire functions of finite rating, rays are effectively constructed, along which the function is unbounded. It is shown that there exists an entire analytic generalized function of infinite rating that is bounded on every ray. The obtained results, in a somewhat modified form, allow for extension to sufficiently wide classes of elliptic systems on the complex plane.