Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 34, 2009, 47-67

BELTRAMI OPERATORS, NON-SYMMETRIC ELLIPTIC EQUATIONS AND QUANTITATIVE JACOBIAN BOUNDS

Giovanni Alessandrini and Vincenzo Nesi

Università di Trieste, Dipartimento di Matematica e Informatica
Via Valerio 12/b, 34100 Trieste, Italia; alessang 'at' units.it

La Sapienza, Università di Roma, Dipartimento di Matematica
P. le A. Moro 2, 00185 Roma, Italia; nesi 'at' mat.uniroma1.it

Abstract. In recent studies on the G-convergence of Beltrami operators, a number of issues arouse concerning injectivity properties of families of quasiconformal mappings. Bojarski, D'Onofrio, Iwaniec and Sbordone formulated a conjecture based on the existence of a so-called primary pair. Very recently, Bojarski 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. 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 C onto itself.

2000 Mathematics Subject Classification: Primary 30C62, 35J55.

Key words: Beltrami operators, quasiconformal mappings.

Reference to this article: G. Alessandrini and V. Nesi: Beltrami operators, non-symmetric elliptic equations and quantitative Jacobian bounds. Ann. Acad. Sci. Fenn. Math. 34 (2009), 47-67.

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