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 $R\subseteq ℂ$. 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

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