On strong solutions of the Beltrami equations

Abstract

We formulate general principles on the existence of homeomorphic absolutely continuous on lines (ACL) solutions for the Beltrami equations with degeneration and derive from them a series of criteria and, in particular, a generalization and strengthening of the well-known Lehto existence theorem. Furthermore, we prove that in all these cases there exist the so-called strong ring solutions satisfying additional moduli conditions which play a great role in the research of various properties of such solutions.

Keywords:

• Beltrami equations
• strong ring solutions

AMS Subject Classifications::

• primary 30C65
• secondary 30C75

Acknowledgements

The research of the first author was partially supported by grants from the University of Helsinki, from Technion–Israel Institute of Technology, Haifa, Holon Institute of Technology, Israel, from Institute of Mathematics of PAN, Warsaw, Poland and by Grant F25.1/055 of the State Foundation of Fundamental Investigations of Ukraine. The research of the second author was partially supported by a grant from the Israel Science Foundation (grant no. 198/00) and by Technion Fund for the Promotion of Research, and the third author was partially supported by a grant from the Israel Science Foundation (grant no. 198/00).