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Search: MSC category 35N15
( $\overline\partial$Neumann problem and generalizations; formal complexes [See also 32W05, 32W10, 58J10] )
1. CJM 1997 (vol 49 pp. 1299)
 Tie, Jingzhi

The explicit solution of the $\bar\partial$Neumann problem in a nonisotropic Siegel domain
In this paper, we solve the $\dbar$Neumann problem
on $(0,q)$ forms, $0\leq q \leq n$, in the strictly
pseudoconvex nonisotropic Siegel domain:
\[
\cU=\left\{
\begin{array}{clc}
&\bz=(z_1,\ldots,z_n) \in \C^{n},\\
(\bz,z_{n+1}):&&\Im (z_{n+1}) > \sum_{j=1}^{n}a_j z_j^2 \\
&z_{n+1}\in \C;
\end{array}
\right\},
\]
where $a_j> 0$ for $j=1,2,\ldots, n$. The metric we
use is invariant under the action of the Heisenberg
group on the domain. The fundamental solution of the
related differential equation is derived via the
Laguerre calculus. We obtain an explicit formula for
the kernel of the Neumann operator. We also construct
the solution of the corresponding heat equation and
the fundamental solution of the Laplacian operator
on the Heisenberg group.
Categories:32F15, 32F20, 35N15 
