51. CJM 2002 (vol 54 pp. 945)
 Boivin, André; Gauthier, Paul M.; Paramonov, Petr V.

Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications
Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jetdistributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
Keywords:approximation on closed sets, elliptic operator, strongly elliptic operator, $L$meromorphic and $L$analytic functions, localization operator, Banach space of distributions, Dirichlet problem Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30 

52. CJM 2002 (vol 54 pp. 1065)
 Hayashi, Nakao; Naumkin, Pavel I.

Large Time Behavior for the Cubic Nonlinear SchrÃ¶dinger Equation
We consider the Cauchy problem for the cubic nonlinear Schr\"odinger
equation in one space dimension
\begin{equation}
\begin{cases}
iu_t + \frac12 u_{xx} + \bar{u}^3 = 0,
& \text{$t \in \mathbf{R}$, $x \in \mathbf{R}$,} \\
u(0,x) = u_0(x), & \text{$x \in \mathbf{R}$.}
\end{cases}
\label{A}
\end{equation}
Cubic type nonlinearities in one space dimension heuristically appear
to be critical for large time. We study the global existence and
large time asymptotic behavior of solutions to the Cauchy problem
(\ref{A}). We prove that if the initial data $u_0 \in
\mathbf{H}^{1,0} \cap \mathbf{H}^{0,1}$ are small and such that
$\sup_{\xi\leq 1} \arg \mathcal{F} u_0 (\xi)  \frac{\pi n}{2}
< \frac{\pi}{8}$ for some $n \in \mathbf{Z}$, and $\inf_{\xi\leq
1} \mathcal{F} u_0 (\xi) >0$, then the solution has an additional
logarithmic timedecay in the short range region $x \leq
\sqrt{t}$. In the far region $x > \sqrt{t}$ the asymptotics have
a quasilinear character.
Category:35Q55 

53. CJM 2002 (vol 54 pp. 998)
 Dimassi, Mouez

Resonances for Slowly Varying Perturbations of a Periodic SchrÃ¶dinger Operator
We study the resonances of the operator $P(h) = \Delta_x + V(x) +
\varphi(hx)$. Here $V$ is a periodic potential, $\varphi$ a
decreasing perturbation and $h$ a small positive constant. We prove
the existence of shape resonances near the edges of the spectral bands
of $P_0 = \Delta_x + V(x)$, and we give its asymptotic expansions in
powers of $h^{\frac12}$.
Categories:35P99, 47A60, 47A40 

54. CJM 2002 (vol 54 pp. 493)
 Braden, Tom

Perverse Sheaves on Grassmannians
We compute the category of perverse sheaves on Hermitian symmetric
spaces in types~A and D, constructible with respect to the Schubert
stratification. The calculation is microlocal, and uses the action of
the Borel group to study the geometry of the conormal variety
$\Lambda$.
Keywords:perverse sheaves, microlocal geometry Categories:32S60, 32C38, 35A27 

55. CJM 2001 (vol 53 pp. 1057)
 Varopoulos, N. Th.

Potential Theory in Lipschitz Domains
We prove comparison theorems for the probability of life in a
Lipschitz domain between Brownian motion and random walks.
On donne des th\'eor\`emes de comparaison pour la probabilit\'e de
vie dans un domain Lipschitzien entre le Brownien et de marches
al\'eatoires.
Categories:39A70, 3502, 65M06 

56. CJM 2001 (vol 53 pp. 278)
 Helminck, G. F.; van de Leur, J. W.

Darboux Transformations for the KP Hierarchy in the SegalWilson Setting
In this paper it is shown that inclusions inside the SegalWilson
Grassmannian give rise to Darboux transformations between the
solutions of the $\KP$ hierarchy corresponding to these planes. We
present a closed form of the operators that procure the transformation
and express them in the related geometric data. Further the
associated transformation on the level of $\tau$functions is given.
Keywords:KP hierarchy, Darboux transformation, Grassmann manifold Categories:22E65, 22E70, 35Q53, 35Q58, 58B25 

57. CJM 2000 (vol 52 pp. 757)
 Hanani, Abdellah

Le problÃ¨me de Neumann pour certaines Ã©quations du type de MongeAmpÃ¨re sur une variÃ©tÃ© riemannienne
Let $(M_n,g)$ be a strictly convex riemannian manifold with
$C^{\infty}$ boundary. We prove the existence\break
of classical solution for the nonlinear elliptic partial
differential equation of MongeAmp\`ere:\break
$\det (u\delta^i_j + \nabla^i_ju) = F(x,\nabla u;u)$ in $M$ with a
Neumann condition on the boundary of the form $\frac{\partial
u}{\partial \nu} = \varphi (x,u)$, where $F \in C^{\infty} (TM
\times \bbR)$ is an everywhere strictly positive function
satisfying some assumptions, $\nu$ stands for the unit normal
vector field and $\varphi \in C^{\infty} (\partial M \times \bbR)$
is a nondecreasing function in $u$.
Keywords:connexion de LeviCivita, Ã©quations de MongeAmpÃ¨re, problÃ¨me de Neumann, estimÃ©es a priori, mÃ©thode de continuitÃ© Categories:35J60, 53C55, 58G30 

58. CJM 2000 (vol 52 pp. 522)
 Gui, Changfeng; Wei, Juncheng

On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems
We consider the problem
\begin{equation*}
\begin{cases}
\varepsilon^2 \Delta u  u + f(u) = 0, u > 0 & \mbox{in } \Omega\\
\frac{\partial u}{\partial \nu} = 0 & \mbox{on } \partial\Omega,
\end{cases}
\end{equation*}
where $\Omega$ is a bounded smooth domain in $R^N$, $\ve>0$ is a small
parameter and $f$ is a superlinear, subcritical nonlinearity. It is
known that this equation possesses multiple boundary spike solutions
that concentrate, as $\epsilon$ approaches zero, at multiple critical
points of the mean curvature function $H(P)$, $P \in \partial \Omega$.
It is also proved that this equation has multiple interior spike solutions
which concentrate, as $\ep\to 0$, at {\it sphere packing\/} points in $\Om$.
In this paper, we prove the existence of solutions with multiple spikes
{\it both\/} on the boundary and in the interior. The main difficulty
lies in the fact that the boundary spikes and the interior spikes usually
have different scales of error estimation. We have to choose a special set
of boundary spikes to match the scale of the interior spikes in a
variational approach.
Keywords:mixed multiple spikes, nonlinear elliptic equations Categories:35B40, 35B45, 35J40 

59. CJM 2000 (vol 52 pp. 119)
60. CJM 1999 (vol 51 pp. 372)
 Mytnik, Leonid

Uniqueness for a Competing Species Model
We show that a martingale problem associated with a competing
species model has a unique solution. The proof of uniqueness of the
solution for the martingale problem is based on duality
technique. It requires the construction of dual probability
measures.
Keywords:stochastic partial differential equation, Martingale problem, duality Categories:60H15, 35R60 

61. CJM 1998 (vol 50 pp. 1119)
 Anand, Christopher Kumar

Ward's solitons II: exact solutions
In a previous paper, we gave a correspondence between certain exact
solutions to a \((2+1)\)dimensional integrable Chiral Model and
holomorphic bundles on a compact surface. In this paper, we use
algebraic geometry to derive a closedform expression for those
solutions and show by way of examples how the algebraic data which
parametrise the solution space dictates the behaviour of the
solutions.
Dans un article pr\'{e}c\'{e}dent, nous avons d\'{e}montr\'{e} que
les solutions d'un mod\`{e}le chiral int\'{e}grable en dimension \(
(2+1) \) correspondent aux fibr\'{e}s vectoriels holomorphes sur
une surface compacte. Ici, nous employons la g\'{e}om\'{e}trie
alg\'{e}brique dans une construction explicite des solutions. Nous
donnons une formule matricielle et illustrons avec trois exemples
la signification des invariants alg\'{e}briques pour le
comportement physique des solutions.
Keywords:integrable system, chiral field, sigma model, soliton, monad, uniton, harmonic map Category:35Q51 

62. CJM 1998 (vol 50 pp. 1298)
 Milson, Robert

Imprimitively generated Liealgebraic Hamiltonians and separation of variables
Turbiner's conjecture posits that a Liealgebraic Hamiltonian
operator whose domain is a subset of the Euclidean plane admits a
separation of variables. A proof of this conjecture is given in
those cases where the generating Liealgebra acts imprimitively.
The general form of the conjecture is false. A counterexample is
given based on the trigonometric OlshanetskyPerelomov potential
corresponding to the $A_2$ root system.
Categories:35Q40, 53C30, 81R05 

63. CJM 1998 (vol 50 pp. 1090)
64. CJM 1998 (vol 50 pp. 605)
65. CJM 1998 (vol 50 pp. 487)
 Barlow, Martin T.

On the Liouville property for divergence form operators
In this paper we construct a bounded strictly positive
function $\sigma$ such that the Liouville property fails for the
divergence form operator $L=\nabla (\sigma^2 \nabla)$. Since in
addition $\Delta \sigma/\sigma$ is bounded, this example also gives a
negative answer to a problem of Berestycki, Caffarelli and Nirenberg
concerning linear Schr\"odinger operators.
Categories:31C05, 60H10, 35J10 

66. CJM 1998 (vol 50 pp. 40)
 Engliš, Miroslav; Peetre, Jaak

Green's functions for powers of the invariant Laplacian
The aim of the present paper is the computation of Green's functions
for the powers $\DDelta^m$ of the invariant Laplace operator on rankone
Hermitian symmetric spaces. Starting with the noncompact case, the
unit ball in $\CC^d$, we obtain a complete result for $m=1,2$ in
all dimensions. For $m\ge3$ the formulas grow quite complicated so
we restrict ourselves to the case of the unit disc ($d=1$) where
we develop a method, possibly applicable also in other situations,
for reducing the number of integrations by half, and use it to give
a description of the boundary behaviour of these Green functions
and to obtain their (multivalued) analytic continuation to the
entire complex plane. Next we discuss the type of special functions
that turn up (hyperlogarithms of Kummer). Finally we treat also
the compact case of the complex projective space $\Bbb P^d$ (for
$d=1$, the Riemann sphere) and, as an application of our results,
use eigenfunction expansions to obtain some new identities involving
sums of Legendre ($d=1$) or Jacobi ($d>1$) polynomials and the
polylogarithm function. The case of Green's functions of powers of
weighted (no longer invariant, but only covariant) Laplacians is
also briefly discussed.
Keywords:Invariant Laplacian, Green's functions, dilogarithm, trilogarithm, Legendre and Jacobi polynomials, hyperlogarithms Categories:35C05, 33E30, 33C45, 34B27, 35J40 

67. 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 

68. CJM 1997 (vol 49 pp. 798)
69. CJM 1997 (vol 49 pp. 232)
 Edward, Julian

Spectral theory for the Neumann Laplacian on planar domains with hornlike ends
The spectral theory for the Neumann Laplacian on planar domains with
symmetric, hornlike ends is studied. For a large class of such domains,
it is proven that the Neumann Laplacian has no singular continuous
spectrum, and that the pure point spectrum consists of eigenvalues
of finite multiplicity which can accumulate only at $0$ or $\infty$.
The proof uses Mourre theory.
Categories:35P25, 58G25 
