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1. CJM 2005 (vol 57 pp. 1279)
A Semilinear Problem for the Heisenberg Laplacian on Unbounded Domains We study the semilinear equation
\begin{equation*}
-\Delta_{\mathbb H} u(\eta) + u(\eta) = f(\eta,
u(\eta)),\quad
u \in \So(\Omega),
\end{equation*}
where $\Omega$ is an unbounded domain of the Heisenberg
group $\mathbb H^N$, $N\ge 1$. The space $\So(\Omega)$ is the
Heisenberg analogue of the Sobolev space $W_0^{1,2}(\Omega)$.
The function $f\colon \overline{\Omega}\times
\mathbb R\to \mathbb R$ is supposed to be odd in $u$,
continuous and satisfy some (superlinear but subcritical) growth
conditions. The operator $\Delta_{\mathbb H}$ is
the subelliptic Laplacian on the Heisenberg group. We
give a condition on $\Omega$ which implies the existence of
infinitely many solutions of the above equation. In the proof we
rewrite the equation as a variational problem, and show that the
corresponding functional satisfies the Palais--Smale
condition. This might be quite surprising since we deal with
domains which are far from bounded. The technique we use rests on
a compactness argument and the maximum principle.
Keywords:Heisenberg group, concentration compactness, Heisenberg Laplacian Categories:22E30, 22E27 |
2. CJM 2005 (vol 57 pp. 598)
Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group Differential operators $D_x$, $D_y$, and $D_z$ are formed using the
action of the $3$-dimensional discrete Heisenberg group $G$ on a set
$S$, and the operators will act on functions on $S$. The Laplacian
operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with
variable differences which can be associated to a unitary
representation of $G$ on the Hilbert space $L^2(S)$. Using techniques
from harmonic analysis and representation theory, we show that the
Laplacian operator is locally solvable.
Keywords:discrete Heisenberg group,, unitary representation,, local solvability,, difference operator Categories:43A85, 22D10, 39A70 |
3. CJM 2004 (vol 56 pp. 566)
Geodesics in a Manifold with Heisenberg Group as Boundary The Heisenberg group is considered as the boundary of a manifold. A class
of hypersurfaces in this manifold can be regarded as copies of the Heisenberg
group. The properties of geodesics in the interior and on the hypersurfaces
are worked out in detail. These properties are strongly related to those of
the Heisenberg group.
Keywords:Heisenberg group, Hamiltonian mechanics, geodesic Categories:53C22, 53C17 |