# Difference between revisions of "Neumann d-bar problem"

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].