Uniqueness of normalized homeomorphic solutions to nonlinear Beltrami equations

TL;DR

This paper investigates the conditions under which normalized homeomorphic solutions to nonlinear Beltrami equations are unique, establishing that uniqueness depends on explicit bounds on ellipticity at infinity.

Contribution

The paper provides a clear criterion for the uniqueness of solutions based on ellipticity bounds, clarifying when uniqueness holds for nonlinear Beltrami equations.

Findings

Uniqueness holds under explicit ellipticity bounds at infinity.

Uniqueness does not hold in general without these bounds.

The results clarify the conditions for solution uniqueness in nonlinear Beltrami equations.

Abstract

We settle the problem of the uniqueness of normalized homeomorphic solutions to nonlinear Beltrami equations $\bar\partial f(z)=H(z, \partial f(z))$. It turns out that the uniqueness holds under definite and explicit bounds on the ellipticity at infinity, but not in general.