Annales Academię Scientiarum Fennicę
Mathematica
Volumen 35, 2010, 59-73
Instytut Matematyczny Polskiej Akademii Nauk
ul. Sniadeckich 8, 00-956 Warszawa, Poland; B.Bojarski 'at' impan.gov.pl
Abstract. We prove that the quasiregular mappings given by the (normalized) principal solution of the linear Beltrami equation (1) and the principal solution of the quasilinear Beltrami equation are inverse to each other. This basic fact is deduced from the Liouville theorem for generalized analytic functions. It essentially simplifies the known proofs of the "measurable Riemann mapping theorem" and its holomorphic dependence on parameters.
2000 Mathematics Subject Classification: Primary 30C62; Secondary 35J55.
Key words: Beltrami operators, quasiconformal mappings, elliptic PDE's.
Reference to this article: B. Bojarski: On the Beltrami equation, once again: 54 years later. Ann. Acad. Sci. Fenn. Math. 35 (2010), 59-73.
doi:10.5186/aasfm.2010.3504
Copyright © 2010 by Academia Scientiarum Fennica