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''Neumann DBAR problem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n0664103.png" />-problem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n0664105.png" />-Neumann problem, DBAR problem, Neumann problem for the Cauchy–Riemann complex''
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{{TEX|done}}
  
A non-coercive boundary problem for the complex Laplacian. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n0664106.png" /> be a relatively compact domain of a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n0664107.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n0664108.png" /> with smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n0664109.png" />. The Cauchy–Riemann operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641010.png" /> (defined on functions on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641011.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641012.png" />) naturally extends to define the Dolbeault complex or Cauchy–Riemann complex
+
''Neumann DBAR problem,  $  \overline \partial \; $-
 +
problem,  $  \overline \partial \; $-
 +
Neumann problem, DBAR problem, Neumann problem for the Cauchy–Riemann complex''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641013.png" /></td> </tr></table>
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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]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641014.png" /> is the space of differential forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641016.png" />. The holomorphic functions are the solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641017.png" /> and the inhomogeneous equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641018.png" /> (under the necessary compactibility condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641019.png" />) is also of interest. For instance, in connection with the Levi problem: Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641020.png" />, is there a holomorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641021.png" /> which blows up at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641022.png" />? Using a general formalism of D.C. Spencer (and general Hilbert space theory), the problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641023.png" /> leads to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641024.png" />-Neumann problem
+
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 ,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$
 
+
where $  u \in \Lambda ^{p,q+1} (M) $,  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641026.png" /> is the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641027.png" />, which is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641028.png" />, where the inner product is given by integration with respect to the volume form determined by a given Hermitian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641029.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641030.png" /> is called the complex Laplacian. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641031.png" /> is a [[Kähler manifold|Kähler manifold]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641033.png" /> is the usual Laplacian of the de Rham complex, cf. [[De Rham cohomology|de Rham cohomology]].
+
$  \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) $;
Strictly speaking, equation (a1) should be written as
+
$  q = - 1 ,\  0 \dots n + 1 $,  
 
+
$  \Lambda ^{p,-1} (M) = 0 = \Lambda ^{p,n+2} (M) $.  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
Thus equation (A2) comes naturally equipped with the boundary conditions $$ \tag{A3}
 
+
u  \in  \textrm{ Domain }  ( \overline \partial \; {} _{q} ^{*} ),
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641037.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641039.png" />. Thus equation (a2) comes naturally equipped with the boundary conditions
+
$$
 
+
$$ \tag{A4}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\overline \partial \; _{q+1} u  \in  \textrm{ Domain }  ( \overline \partial \; {} _{q+1} ^{*} ).
 
+
$$(
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641041.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
The $  \overline \partial \; $-
 
+
Neumann boundary conditions.) The operator $  \square $
(The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641043.png" />-Neumann boundary conditions.) The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641044.png" /> 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
+
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 \| ,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641045.png" /></td> </tr></table>
+
$$
 
+
where  $  \| \cdot \| _{s} $
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066410/n06641046.png" /> 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]]].
+
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====
 
====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>
 
<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>

Latest revision as of 11:21, 22 December 2019


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, then $ \square = \Delta /2 $, where $ \Delta $ is the usual Laplacian of the de Rham complex, cf. 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). For more details cf. [a1], [a2]. A great deal of additional and related material can be found in [a1][a4].

References

[a1] G.B. Folland, J.J. Kohn, "The Neumann problem for the Cauchy–Riemann complex" , Annals Math. Studies , 75 , Princeton Univ. Press (1972)
[a2] P.C. Greiner, E.M. Sfein, "Estimates for the -Neumann problem" , Princeton Univ. Press (1977)
[a3] F. Trèves, "Introduction to pseudodifferential and Fourier integral operators" , 1 , Plenum (1980) pp. Sect. III.8
[a4] 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
How to Cite This Entry:
Neumann d-bar problem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Neumann_d-bar_problem&oldid=44321