On the Beltrami Equation, Once Again: 54 Years Later

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. The first global, i.e. defined in the full complex plane C and expressed by an explicit analytical formula, solution of the Beltrami equation (1) wz̄ − q(z)wz = 0 was given by Vekua in the years 1953–54 and it appeared in the first issue of Doklady for 1955 [32]. Vekua in [32] considered the equation (1) with compactly supported q(z), q(z) ≡ 0 for |z| > R, for some finite R, satisfying the uniform ellipticity condition (2) |q(z)| ≤ q0 < 1, q0 − const. In [32] he considers the class of solutions of (1) represented by the Cauchy complex potential Tω in the form