Semilinear equations in a plane and quasiconformal mappings

1Gutlyanskii, VYa., 1Nesmelova, OV, 1Ryazanov, VI
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
Dopov. Nac. akad. nauk Ukr. 2017, 1:10-16
Section: Mathematics
Language: English

We consider generalizations of the Bieberbach equation with nonlinear right parts, which makes it possible to study many problems of mathematical physics in inhomogeneous and anisotropic media with smooth characteristics. We establish interconnections of these semilinear equations with quasiconformal mappings, obtain on this basis, a series of theorems on the existence of their solutions that blow-up on the boundary of a unit disk, as well as on punctured unit disks and rings, and give their explicit representations.

Keywords: Beltrami equation, Bieberbach equation, Keller– Osserman condition, quasiconformal mappings, semilinear elliptic equations
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