Nonlinear Cauchy-Riemann operators in

Author:
Tadeusz Iwaniec

Journal:
Trans. Amer. Math. Soc. **354** (2002), 1961-1995

MSC (2000):
Primary 35J60, 30G62; Secondary 42B25, 26B10

DOI:
https://doi.org/10.1090/S0002-9947-02-02914-8

Published electronically:
January 8, 2002

MathSciNet review:
1881026

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper has arisen from an effort to provide a comprehensive and unifying development of the -theory of quasiconformal mappings in . The governing equations for these mappings form nonlinear differential systems of the first order, analogous in many respects to the Cauchy-Riemann equations in the complex plane. This approach demands that one must work out certain variational integrals involving the Jacobian determinant. Guided by such integrals, we introduce two nonlinear differential operators, denoted by and , which act on weakly differentiable deformations of a domain .

Solutions to the so-called Cauchy-Riemann equations and are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental -estimate

In quest of the best constant , we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.

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Additional Information

**Tadeusz Iwaniec**

Affiliation:
Department of Mathematics, Syracuse University, Syracuse, New York 13244

Email:
tiwaniec@mailbox.syr.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-02914-8

Keywords:
Jacobians,
sharp estimates for singular integrals,
rank-one convexity,
quasiconformal mappings

Received by editor(s):
October 10, 1998

Published electronically:
January 8, 2002

Additional Notes:
Supported in part by NSF grant DMS-9706611

Article copyright:
© Copyright 2002
American Mathematical Society