Annales Academię Scientiarum Fennicę

Mathematica

Volumen 35, 2010, 59-73

# ON THE BELTRAMI EQUATION, ONCE AGAIN: 54 YEARS LATER

## Bogdan Bojarski

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.

Full document as PDF file

doi:10.5186/aasfm.2010.3504

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