View source for Neumann d-bar problem

''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''

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

<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>

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

<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>

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]].

Strictly speaking, equation (a1) should be written as

<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>

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

<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>

<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 <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

<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 <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]]].

====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>