# Beltrami equation with harmonic coefficient

I need to find solutions to the Beltrami equation

$$\frac{\partial w}{\partial\overline{{z}}}=e^{i\varphi(z)}\frac{\partial w}{\partial z}$$

for $$w=w(z,\overline{z})$$ and $$\varphi(z)$$ some given, real, harmonic function. So the Beltrami coefficient is just a phase.

Among the almost infinite literature about the Beltrami equation the only thing I've been able to find is that the solution exists and is unique, provided that $$w$$ is specified along a compact contour on the plane. Solving it numerically is possible, but I'd like to have a deeper understanding of the solutions, and in particular, what are the implications of $$\varphi(z)$$ being harmonic, and even more, of the Beltrami coefficient being a phase, $$|\mu(z)|=1$$, a fact that certainty drives the problem.

Any ideas ?

• Your equation might have a solution but it will not be quasiconformal (I think, its image will be a curve). For quasiconformality you need a Beltrami differential $\mu$ satisfying $||\mu||_\infty<1$. Commented Dec 29, 2020 at 21:45
• $\|\mu\|_\infty=1$ implies that $w$ is everywhere degenerate, $J_w\equiv 0$. Commented Dec 29, 2020 at 23:57

Note that, if you take $$\phi=0$$, then the equation reduces to $$w_y =0$$, i.e., if $$D\subset C$$ is the domain of $$w$$ and $$x:D\to\mathbb{R}$$ is the projection on the $$x$$-axis and has connected fibers, then $$w= h(x)$$ for some $$C^1$$ function $$h:x(D)\to\mathbb{C}$$, and this is the general solution on such $$D$$.
Something similar happens in general: Write $$\mathrm{d}w = w_z\,\mathrm{d}z + w_{\bar z}\,\mathrm{d}\bar z = w_z\,(\mathrm{d}z + \mathrm{e}^{i\phi(z)}\,\mathrm{d}\bar z) = \mathrm{e}^{i\phi(z)/2}w_z\left(\mathrm{e}^{-i\phi(z)/2}\mathrm{d}z + \mathrm{e}^{i\phi(z)/2}\,\mathrm{d}\bar z\right).$$ Then, setting $$\alpha = \mathrm{e}^{-i\phi(z)/2}\mathrm{d}z + \mathrm{e}^{i\phi(z)/2}\,\mathrm{d}\bar z$$, we see that $$\alpha$$ is a real-valued $$1$$-form, and hence always has a local integrating factor, i.e., it can be written locally in the form $$\alpha = f\,\mathrm{d}u$$ for some functions real-valued functions $$u$$ and $$f>0$$. Thus, if $$D\subset\mathbb{C}$$ is a domain such that $$\alpha$$ can be written as $$\alpha = f\,\mathrm{d}u$$ for some real-valued functions $$u$$ and $$f>0$$ on $$D$$ and the fibers of $$u:D\to u(D)\subset \mathbb{R}$$ are connected, then any solution of your equation on $$D$$ can be written in the form $$w = h(u)$$ for some $$C^1$$ function $$h:u(D)\to\mathbb{C}$$, and every such $$h$$ that is $$C^1$$ yields a solution. This is because your equation for $$w:D\to\mathbb{C}$$ reduces to $$\mathrm{d}w = p\,\mathrm{d}u$$ for some function $$p:D\to\mathbb{C}$$.
The signficance of $$\phi$$ being harmonic is not really clear (other than ensuring that $$\alpha$$ is real-analytic, so that $$u$$ can be taken to be real-analytic also). Certainly, the behavior of $$\phi$$ will determine which domains $$D\subset\mathbb{C}$$ have the right shape to support an integrating factor for $$\alpha$$, but it is not clear to me that just requiring that $$\phi$$ be harmonic gives you much easily accessible information along those lines.