Manifolds of quasiconformal mappings and the nonlinear Beltrami equation

Abstract

In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation ${\partial }_{\bar{z}}f=\mathcal{H}(z,{\partial }_{z}f)$ generate a two-dimensional manifold of quasiconformal mappings ${\mathcal{F}}_{\mathcal{H}}\subset W_{\mathrm{loc}}^{1,2}(\mathbb{C})$. Moreover, we show that under regularity assumptions on $\mathcal{H}$, the manifold ${\mathcal{F}}_{\mathcal{H}}$ defines the structure function $\mathcal{H}$ uniquely.

26 pages, Proposition 2.2 is improved and the proofs of Schauder estimates and the lower bound for the Jacobian are moved to arXiv:1511.08370

References in corpus (1)

• Nonlinear Beltrami operators, Schauder estimates and bounds for the Jacobian

Cited by in corpus (1)

• Improved Hölder regularity for strongly elliptic PDEs