# Degenerate Beltrami equation

Question:
Let $$\mu:\mathbb C\to \mathbb C$$ be a $$C^\infty$$ function satisfying $$|\mu|\le 1$$.
Let us furthermore assume that the function $$\mu$$ never takes the value $$-1$$.
Does there exist a $$C^\infty$$ function

$$f:\mathbb C\to \mathbb C$$ such that $$\frac{\,\partial f/\partial \bar z\,}{\partial f/\partial z}=\mu$$ ?
(and $$\scriptstyle\partial f/\partial z$$ everywhere non-zero)

Note that $$f$$ is by no means expected to be a homeomorphism. For example, if $$\mu\equiv 1$$, then a general solution is of the form projection $$\mathbb C\to\mathbb R:z\mapsto Re(z)$$ followed by any embedding $$\mathbb R\to\mathbb C$$.

The special case $$|\mu|<1$$ is the usual Beltrami equation.
More generally, on which domains $$U\subset\mathbb C$$ can one guarantee that a solution exists?

[The exact problem I care about is defined on the strip $$U=\{z\in\mathbb C:0\le Im(z)\le 1\}$$, and the function $$f$$ is furthermore required to be periodic: $$f(z)=f(z+1)$$.]

Equivalent formulation:
Let $$\nu:\mathbb C\to \mathbb C$$ be a $$C^\infty$$ function satisfying $$Im(\nu)\ge0$$.

$$\exists$$? $$f:\mathbb C\to \mathbb C$$ such that $$\frac{\,\partial f/\partial y\,}{\partial f/\partial x}=\nu$$ ?
(and $$\scriptstyle\partial f/\partial x$$ everywhere non-zero)

(The two formulations are equivalent via the transformations $$\mu=\frac{1+i\nu}{1-i\nu}$$ and $$\nu=i\frac{1-\mu}{1+\mu}$$, which are a version of the Cayley transform. These transformations map $$\nu$$ in the closed upper half plane to $$\mu$$ in the closed unit disc minus $$-1$$.)

Partial result (positive solution in the completely degenerate case):
When $$\nu$$ is real, equivalently when $$|\mu|=1$$, then the problem admits a positive solution.

In that case, the vector field $$\partial_y-\nu\partial_x$$ defines a foliation of $$\mathbb C$$. That vector field is never horizontal, so all the leaves of the foliation are closed (diffeomorphic to $$\mathbb R$$). Any function $$f:\mathbb C\to\mathbb C$$ which is constant on the leaves (i.e., which factors through the 1-dimensional leaf space) is then a solution of the degenerate Beltrami equation $$\frac{\,\partial f/\partial y\,}{\partial f/\partial x}=\nu$$.

Note that, unlike the leaf space of a general foliation of $$\mathbb C$$, the leaf space of this specific foliation is a manifold (by which I mean a second-countable possibly non-Hausdorff manifold), and can thus be immersed into $$\mathbb C$$. That's where the crucial condition $$\mu\not=-1$$ gets used.

A reference for these types of equations is in Astala, Iwaniec and Martin chapter 20. One of the main difficulties is showing that $$f$$ is a homeomorphism. In general it will not be the case. For example $$f(z) = x$$ has a Beltrami coefficient $$\mu \equiv 1$$ but it sends the entire plane into the real line.
• Thank you for the reference; I will have a look. In my cases of interest, $f$ will typically NOT be a homeomorphism. These are exactly the cases I care about. Sep 13, 2019 at 16:16