generalization of Cauchy-Riemann conditions

Question

Esteemed experts,

Please excuse the ignorance and language of a poor physicist.

1. As we know, the real $u(x,y)$ and imaginary $v(x,y)$ parts of an analytic function (in some domain) satisfy the Cauchy-Riemann (CR) conditions

   $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},$

   $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.$

2. In contrast, in the problem that I encountered, I am looking at the following equations

   $\frac{\partial u}{\partial x} = \sigma(x,y)\,\frac{\partial v}{\partial y},$

   $\frac{\partial u}{\partial y} = -\sigma(x,y)\,\frac{\partial v}{\partial x},$

   where $\sigma(x,y)$ is a sufficiently smooth function. The latter equations look like the CR conditions but are not equivalent to them because $\sigma(x,y)$ depends on $(x,y)$.

3. My question is whether it is possible to generalize the complex analysis to have the generalized Cauchy-Riemann conditions with position-dependent $\sigma(x,y)$. Perhaps someone can point me to the relevant references.

Thanks in advance, Serge

Answer 1

The equation that you have is a special case of the more general class of quasilinear differential equations in the complex plane studied by Bojarskii (and others); the English translation of his paper can be found here,

Bojarski, B. V., Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients. Translated from the 1957 Russian original. With a foreword by Eero Saksman, Report. University of Jyväskylä. Department of Mathematics and Statistics 118. Jyväskylä: University of Jyväskylä, Department of Mathematics and Statistics (ISBN 978-951-39-3486-6/pbk). 64 p. (2009). ZBL1173.35403.

In the context of your problem, Bojarski considers a measurable (more precisely, $L_\infty$) function $\sigma$ of two variables in an (open, connected and bounded) domain $D$ in the complex plane. He proves the existence of solution (section 2 of the paper) and much more (including a form of uniqueness, see section 6 of his paper). Solutions in his case are understood in the sense of distributions (he would allow discontinuous functions $\sigma$) under the very mild assumption that $|\sigma|$ is bounded away from zero in $D$. Since you probably do not know what distributions are, just assume continuity of $\sigma$. If and when you try to read his paper, simply replace "measurable" with "continuous" and then you can work with classical (instead of distributional) derivatives.

The solutions of this equation are generalizations of holomorphic functions (sometimes called "quasiregular") and share many of their properties. For instance, "generically" the solution is a local quasiconformal homeomorphism. In general, solutions can be described as compositions $h\circ f$, where $h$ is a holomorphic function and $f$ is a quasiconformal homeomorphism.

The case when $\sigma$ is allowed to approach zero were also subsequently studied, see for instance

Sevost’yanov, E. A., On quasilinear Beltrami-type equations with degeneration, Math. Notes 90, No. 3, 431-438 (2011); translation from Math. Zametki 90, No. 3, 445-453 (2011). ZBL1291.30140.

The nature of solutions in this case is much less clear (at least to me).

Edit. Here is some background on quasiconformal maps.