Let $ a $, $ b $ and $ c $ be real constants such that $ \Delta \stackrel{\text{df}}{=} a c - b^{2} > 0 $. The *Beltrami Equations* are defined as the following system of PDE’s on the domain $ \Bbb{R}^{2} $:
$$
u\_{x} = \frac{1}{\sqrt{\Delta}} (b v\_{x} + c v\_{y}), \qquad
u\_{y} = - \frac{1}{\sqrt{\Delta}} (a v\_{x} + b v\_{y}).
$$
Boundary conditions are not imposed.



> 
> **Question:** Are solutions $ u $ and $ v $ of this system necessarily smooth?
> 
> 
> 


I have been told that the Elliptic Regularity Theorem answers my question, but the theorem applies only to even-order elliptic operators and what we have here are only first-order equations, so we do not know if solutions $ u $ and $ v $ have even second-order partial derivatives. Observe that we have the same scenario for the Cauchy-Riemann Equations (in PDE form).


I understand that my question may be too basic for MathOverflow, so I understand if anyone wishes to close it. Thank you!



Are solutions of the Beltrami Equations necessarily smooth?
As Deane Yang suggested you can regard the first order Beltrami system as an elliptic system, see Chapter 7.3 in the following book


Astala, Kari; Iwaniec, Tadeusz; Martin, Gaven Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009. 


Also Chapter 5 and 6 give the answer for your question. 


Since your equation is linear, you could have a representation formula for your equation, see for instance the following nice paper


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. University of Jyväskylä, Jyväskylä, 2009.


