DBAR problem

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

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

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