kb/data/en.wikipedia.org/wiki/DBAR_problem-0.md

---
title: "DBAR problem"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/DBAR_problem"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:04:45.844723+00:00"
instance: "kb-cron"
---

The DBAR problem, or the 
  
    
      
        
          
            
              ∂
              ¯
            
          
        
      
    
    {\displaystyle {\bar {\partial }}}
  
-problem, is the problem of solving the differential equation

  
    
      
        
          
            
              ∂
              ¯
            
          
        
        f
        (
        z
        ,
        
          
            
              z
              ¯
            
          
        
        )
        =
        g
        (
        z
        )
      
    
    {\displaystyle {\bar {\partial }}f(z,{\bar {z}})=g(z)}
  

for the function 
  
    
      
        f
        (
        z
        ,
        
          
            
              z
              ¯
            
          
        
        )
      
    
    {\displaystyle f(z,{\bar {z}})}
  
, where 
  
    
      
        g
        (
        z
        )
      
    
    {\displaystyle g(z)}
  
 is assumed to be known and 
  
    
      
        z
        =
        x
        +
        i
        y
      
    
    {\displaystyle z=x+iy}
  
 is a complex number in a domain 
  
    
      
        R
        ⊆
        
          C
        
      
    
    {\displaystyle R\subseteq \mathbb {C} }
  
. The operator 
  
    
      
        
          
            
              ∂
              ¯
            
          
        
      
    
    {\displaystyle {\bar {\partial }}}
  
 is called the DBAR operator:

  
    
      
        
          
            
              ∂
              ¯
            
          
        
        =
        
          
            1
            2
          
        
        
          (
          
            
              
                ∂
                
                  ∂
                  x
                
              
            
            +
            i
            
              
                ∂
                
                  ∂
                  y
                
              
            
          
          )
        
      
    
    {\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

  
    
      
        ∂
        =
        
          
            ∂
            
              ∂
              z
            
          
        
        =
        
          
            1
            2
          
        
        
          (
          
            
              
                ∂
                
                  ∂
                  x
                
              
            
            −
            i
            
              
                ∂
                
                  ∂
                  y
                
              
            
          
          )
        
      
    
    {\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 
  
    
      
        z
      
    
    {\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.


== Citations ==


== 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.[1]
Konopelchenko, B. G. (2000). "On dbar-problem and integrable equations". arXiv:nlin/0002049.