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

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{equation*}\|\mathcal {D}^{+}f\|_{p} \le A_{p}(n)\|\mathcal {D}^{-}f\|_{p}. \end{equation*}

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.