Beltrami operators, non-symmetric elliptic equations and quantitative Jacobian bounds. (English) Zbl 1177.30019

Authors’ abstract: In recent studies on the \(G\)-convergence of Beltrami operators a number of issues arose concerning injectivity properties of families of quasiconformal mappings. B. Bojarski, L. D’Onofrio, T. Iwaniec and C. Sbordone [Ric. Mat. 54, No. 2, 403–432 (2005; Zbl 1139.30311)] formulated a conjecture based on the existence of a so-called primary pair. Very recently, B. Bojarski [Ann. Acad. Sci. Fenn., Math. 32, No. 2, 549–557 (2007; Zbl 1128.31003)] proved the existence of one such pair. We provide a general constructive procedure for obtaining a new rich class of such primary pairs. This proof is obtained as a slight adaptation of previous work by the authors concerning the nonvanishing of the Jacobian of pairs of solutions of elliptic equations in divergence form in the plane, see [Arch. Ration. Mech. Anal. 158, No. 2, 155–171 (2001; Zbl 0977.31006)]. It is proven here that the results previously obtained when the coefficient matrix is symmetric also extend to the non-symmetric case. We also prove a much stronger result giving a quantitative bound for the Jacobian determinant of the so-called periodic \( \sigma\)-harmonic sense preserving homeomorphisms of \(\mathbb C\) onto itself.


30C62 Quasiconformal mappings in the complex plane
35J57 Boundary value problems for second-order elliptic systems
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