DBAR problem

The DBAR problem, or the [math]\displaystyle{ \bar{\partial} }[/math]-problem, is the problem of solving the differential equation [math]\displaystyle{ \bar{\partial} f (z, \bar{z}) = g(z) }[/math] for the function [math]\displaystyle{ f(z,\bar{z}) }[/math], where [math]\displaystyle{ g(z) }[/math] is assumed to be known and [math]\displaystyle{ z = x + iy }[/math] is a complex number in a domain [math]\displaystyle{ R\subseteq \Complex }[/math]. The operator [math]\displaystyle{ \bar{\partial} }[/math] is called the DBAR operator [math]\displaystyle{ \bar{\partial} = \frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) }[/math]

The DBAR operator is nothing other than the complex conjugate of the operator [math]\displaystyle{ \partial=\frac{\partial}{\partial z} = \frac{1}{2} \left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) }[/math] denoting the usual differentiation in the complex [math]\displaystyle{ z }[/math]-plane.

The DBAR problem is of key importance in the theory of integrable systems and generalizes the Riemann–Hilbert problem.