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 $R\subseteq \mathbb {C}$. 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 and generalizes the Riemann–Hilbert problem.