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Transactions of the American Mathematical Society

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Nonlinear Cauchy-Riemann operators in $\mathbb{R}^{n}$

Author: Tadeusz Iwaniec
Journal: Trans. Amer. Math. Soc. 354 (2002), 1961-1995
MSC (2000): Primary 35J60, 30G62; Secondary 42B25, 26B10
Published electronically: January 8, 2002
MathSciNet review: 1881026
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Abstract: This paper has arisen from an effort to provide a comprehensive and unifying development of the $L^{p}$-theory of quasiconformal mappings in $\mathbb{R}^{n}$. 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 $\mathcal{D}^{-}$and $\mathcal{D}^{+}$, which act on weakly differentiable deformations $f:\Omega \to \mathbb{R}^{n}$ of a domain $\Omega \subset \mathbb{R}^{n}$.

Solutions to the so-called Cauchy-Riemann equations $\mathcal{D}^{-}f=0$ and $\mathcal{D}^{+}f=0$ 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 $L^{p}$-estimate

\begin{displaymath}\Vert\mathcal{D}^{+}f\Vert _{p} \le A_{p}(n)\Vert\mathcal{D}^{-}f\Vert _{p}. \end{displaymath}

In quest of the best constant $A_{p}(n)$, 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

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