DBAR problem

The DBAR problem, or the $\overline{\partial }$-problem, is the problem of solving the differential equation $$\overline{\partial }f(z,\overline{z})=g(z)$$ for the function $f(z,\overline{z})$, 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 $\overline{\partial }$ is called the DBAR operator: $$\overline{\partial }=\frac{1}{2}(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y})$$ The DBAR operator is nothing other than the complex conjugate of the operator $$\partial =\frac{\partial }{\partial z}=\frac{1}{2}(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y})$$ 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.

References

• Ablowitz, Mark J.; Fokas, A. S. (2003) (in en). Complex Variables: Introduction and Applications. Cambridge University Press. pp. 516,598–626. ISBN 978-0-521-53429-1. https://books.google.com/books?id=SFqbV3i3hO0C. 
• Haslinger, Friedrich (2014) (in en). The d-bar Neumann Problem and Schrödinger Operators. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-031535-6. https://books.google.com/books?id=Em_nBQAAQBAJ. [1]
• Konopelchenko, B. G. (2000). "On dbar-problem and integrable equations". arXiv:nlin/0002049.