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].

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