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, Schrödinger operators and (together with a jump condition) generalizes the Riemann–Hilbert problem.

References

• Ablowitz, Mark J.; Fokas, A. S. (2003). Complex Variables: Introduction and Applications. Cambridge University Press. pp. 516, 598–626. ISBN 978-0-521-53429-1.
• Haslinger, Friedrich (2014). The d-bar Neumann Problem and Schrödinger Operators. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-031535-6.
• Konopelchenko, B. G. (2000). "On dbar-problem and integrable equations". arXiv:nlin/0002049.