DBAR problem

The DBAR problem, or the ${\displaystyle {\bar {\partial }}}$-problem, is the problem of solving the differential equation $${\displaystyle {\bar {\partial }}f(z,{\bar {z}})=g(z)}$$ for the function ${\displaystyle f(z,{\bar {z}})}$, where ${\displaystyle g(z)}$ is assumed to be known and ${\displaystyle z=x+iy}$ is a complex number in a domain ${\displaystyle R\subseteq \mathbb {C} }$. The operator ${\displaystyle {\bar {\partial }}}$ is called the DBAR operator:^[1] $${\displaystyle {\bar {\partial }}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)}$$ The DBAR operator is nothing other than the complex conjugate of the operator $${\displaystyle \partial ={\frac {\partial }{\partial z}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)}$$ denoting the usual differentiation in the complex ${\displaystyle z}$-plane.

The DBAR problem is of key importance in the theory of integrable systems, Schrödinger operators and (together with a jump condition) generalizes the Riemann–Hilbert problem.^[1][2][3]