DBAR problem explained

The DBAR problem, or the

\bar{\partial} -problem, is the problem of solving the differential equation$$\backslash bar\; f\; (z,\; \backslash bar)\; =\; g(z)$$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:$$\backslash bar\; =\; \backslash frac\; \backslash left(\backslash frac\; +\; i\; \backslash frac\; \backslash right)$$

The DBAR operator is nothing other than the complex conjugate of the operator$$\backslash partial=\backslash frac\; =\; \backslash frac\; \backslash left(\backslash frac\; -\; i\; \backslash frac\; \backslash right)$$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.

References

• Book: Ablowitz . Mark J. . Fokas . A. S. . Complex Variables: Introduction and Applications . 2003 . Cambridge University Press . 978-0-521-53429-1 . 516,598-626 . en.
• Book: Haslinger . Friedrich . The d-bar Neumann Problem and Schrödinger Operators . 2014 . Walter de Gruyter GmbH & Co KG . 978-3-11-031535-6 . en. https://www.mat.univie.ac.at/~has/dbar/dbar1.pdf
• Konopelchenko . B. G.. On dbar-problem and integrable equations. 2000 . nlin/0002049.