On an Unboundedness Property of Solutions of Elliptic Systems in the Plane

Grigori Giorgadze (), Giorgi Makatsaria and Nino Manjavidze
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Grigori Giorgadze: Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, Tbilisi 0179, Georgia
Giorgi Makatsaria: Department of Mathematical Cybernetics, Vladimer Chavchanidze Institute of Cybernetics, Tbilisi 0186, Georgia
Nino Manjavidze: Faculty of Business, Technology and Education, Ilia State University, Tbilisi 0179, Georgia

Mathematics, 2025, vol. 13, issue 15, 1-9

Abstract: 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.

Keywords: Carlemann–Bers–Vekua system; generalized analytic function; entire function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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