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''Neumann DBAR problem,  $  \overline \partial \; $-
problem,  $  \overline \partial \; $-
Neumann problem, DBAR problem, Neumann problem for the Cauchy–Riemann complex''

A non-coercive boundary problem for the complex Laplacian. Let  $  M $
be a relatively compact domain of a complex manifold  $  M ^{1} $
of dimension  $  n + 1 $
with smooth boundary  $  b M $. 
The Cauchy–Riemann operator  $  \overline \partial \; $(
defined on functions on a domain  $  M \subset \mathbf C ^{n+1} $
by  $  \overline \partial \; f = \sum _{i=1} ^{n+1} ( {\partial f} / {\partial \overline{z}\; _ i} ) \  d \overline{z}\; _{i} $) 
naturally extends to define the Dolbeault complex or Cauchy–Riemann complex $$ 
0   \rightarrow   \Lambda ^{p,0} (M)    \stackrel{ {\overline \partial \;}} \rightarrow     \Lambda ^{p,1} (M)    \stackrel{ {\overline \partial \;}} \rightarrow   \dots  \stackrel{ {\overline \partial \;}} \rightarrow    
\Lambda ^{p,n+1} (M)   \rightarrow   0 ,
$$
where  $  \Lambda ^{p,q} (M) $
is the space of differential forms of type  $  ( p ,\  q ) $
on  $  M $. 
The holomorphic functions are the solutions of  $  \overline \partial \; f = 0 $
and the inhomogeneous equation  $  \overline \partial \; f = \phi $(
under the necessary compactibility condition  $  \overline \partial \; \phi = 0 $) 
is also of interest. For instance, in connection with the Levi problem: Given  $  x \in b M $, 
is there a holomorphic function on  $  M $
which blows up at  $  x $? 
Using a general formalism of D.C. Spencer (and general Hilbert space theory), the problem  $  \overline \partial \; f = \phi $
leads to the  $  \overline \partial \; $-
Neumann problem $$ \tag{A1}
( \overline \partial \; \overline \partial \; {} ^{*} + \overline \partial \; {} ^{*} \overline \partial \; ) u   =   \phi .
$$
Here  $  \overline \partial \; {} ^{*} $
is the adjoint of  $  \overline \partial \; $, 
which is defined by  $  \langle  \overline \partial \; {} ^{*} f ,\  g \rangle = \langle  f ,\  \overline \partial \; g \rangle $, 
where the inner product is given by integration with respect to the volume form determined by a given Hermitian metric on  $  \overline{M}\; $. 
The operator  $  \square = \overline \partial \; \overline \partial \; {} ^{*} + \overline \partial \; {} ^{*} \overline \partial \; $
is called the complex Laplacian. If  $  M $
is a [[Kähler manifold|Kähler manifold]], then  $  \square = \Delta /2 $, 
where  $  \Delta $
is the usual Laplacian of the de Rham complex, cf. [[De Rham cohomology|de Rham cohomology]].

Strictly speaking, equation (A1) should be written as $$ \tag{A2}
( \overline \partial \; _{q} \overline \partial \; {} _{q} ^{*} + \overline \partial \; {} _{q+1} ^{*} \overline \partial \; _{q+1} ) (u)   =   \phi ,
$$
where  $  u \in \Lambda ^{p,q+1} (M) $, 
$  \overline \partial \; _{q} : \  \Lambda ^{p,q} (M) \rightarrow \Lambda ^{p,q+1} (M) $, 
$  \overline \partial \; {} _{q} ^{*} : \  \Lambda ^{p,q+1} (M) \rightarrow \Lambda ^{p,q} (M) $; 
$  q = - 1 ,\  0 \dots n + 1 $, 
$  \Lambda ^{p,-1} (M) = 0 = \Lambda ^{p,n+2} (M) $. 
Thus equation (A2) comes naturally equipped with the boundary conditions $$ \tag{A3}
u   \in   \textrm{ Domain }  ( \overline \partial \; {} _{q} ^{*} ),
$$
$$ \tag{A4}
\overline \partial \; _{q+1} u   \in   \textrm{ Domain }  ( \overline \partial \; {} _{q+1} ^{*} ).
$$(
The  $  \overline \partial \; $-
Neumann boundary conditions.) The operator  $  \square $
is elliptic, but the boundary conditions are not. Nevertheless, J.J. Kohn was able to prove existence and to provide a systematic analysis of regularity. A main result is the estimate $$ 
\| u \| _{s+1}   \leq   A _{s} \| \square u \| _{s} + \| u \| ,
$$
where  $  \| \cdot \| _{s} $
are Sobolev norms (cf. [[Sobolev space|Sobolev space]]). For more details cf. [[#References|[a1]]], [[#References|[a2]]]. A great deal of additional and related material can be found in [[#References|[a1]]]–[[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.B. Folland,   J.J. Kohn,   "The Neumann problem for the Cauchy–Riemann complex" , ''Annals Math. Studies'' , '''75''' , Princeton Univ. Press  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.C. Greiner,   E.M. Sfein,   "Estimates for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641047.png" />-Neumann problem" , Princeton Univ. Press  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Trèves,   "Introduction to pseudodifferential and Fourier integral operators" , '''1''' , Plenum  (1980)  pp. Sect. III.8</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.J. Kohn,   "Methods of partial differential equations in complex analysis"  R.O. Wells jr. (ed.) , ''Several Complex Variables'' , '''1''' , Amer. Math. Soc.  (1977)  pp. 215–240</TD></TR></table>