Expand all Collapse all | Results 51 - 64 of 64 |
51. CJM 2001 (vol 53 pp. 278)
Darboux Transformations for the KP Hierarchy in the Segal-Wilson Setting In this paper it is shown that inclusions inside the Segal-Wilson
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 |
52. CJM 2000 (vol 52 pp. 757)
Le problÃ¨me de Neumann pour certaines Ã©quations du type de Monge-AmpÃ¨re sur une variÃ©tÃ© riemannienne |
Le problÃ¨me de Neumann pour certaines Ã©quations du type de Monge-AmpÃ¨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 Monge-Amp\`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 non-decreasing function in $u$.
Keywords:connexion de Levi-Civita, Ã©quations de Monge-AmpÃ¨re, problÃ¨me de Neumann, estimÃ©es a priori, mÃ©thode de continuitÃ© Categories:35J60, 53C55, 58G30 |
53. CJM 2000 (vol 52 pp. 522)
On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems |
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 |
54. CJM 2000 (vol 52 pp. 119)
Corrigendum to ``Spectral Theory for the Neumann Laplacian on Planar Domains with Horn-Like Ends'' Errors to a previous paper (Canad. J. Math. (2) {\bf 49}(1997),
232--262) are corrected. A non-standard regularisation of the
auxiliary operator $A$ appearing in Mourre theory is used.
Categories:35P25, 58G25, 47F05 |
55. CJM 1999 (vol 51 pp. 372)
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 |
56. CJM 1998 (vol 50 pp. 1119)
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 closed-form 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 |
57. CJM 1998 (vol 50 pp. 1298)
Imprimitively generated Lie-algebraic Hamiltonians and separation of variables Turbiner's conjecture posits that a Lie-algebraic 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 Lie-algebra acts imprimitively.
The general form of the conjecture is false. A counter-example is
given based on the trigonometric Olshanetsky-Perelomov potential
corresponding to the $A_2$ root system.
Categories:35Q40, 53C30, 81R05 |
58. CJM 1998 (vol 50 pp. 1090)
Sur les transformÃ©es de Riesz sur les groupes de Lie moyennables et sur certains espaces homogÃ¨nes |
Sur les transformÃ©es de Riesz sur les groupes de Lie moyennables et sur certains espaces homogÃ¨nes Let $\Delta$ be a left invariant sub-Laplacian on a Lie group $G$
and let $\nabla$ be the associated gradient. In this paper we
investigate the boundness of the Riesz transform
$\nabla\Delta^{-1/2}$ on Lie groups $G$ which are amenable and with
exponential volume growth and on certain homogenous spaces.
Categories:22E30, 35H05, 43A80, 43A85 |
59. CJM 1998 (vol 50 pp. 605)
Hardy spaces of conjugate systems of temperatures We define Hardy spaces of conjugate systems of temperature functions on
${\bbd R}_{+}^{n+1}$. We show that their boundary distributions are the same
as the boundary distributions of the usual Hardy spaces of conjugate systems
of harmonic functions.
Categories:42B30, 42A50, 35K05 |
60. CJM 1998 (vol 50 pp. 487)
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 |
61. CJM 1998 (vol 50 pp. 40)
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 rank-one
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 (multi-valued) 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 |
62. CJM 1997 (vol 49 pp. 1299)
The explicit solution of the $\bar\partial$-Neumann problem in a non-isotropic Siegel domain In this paper, we solve the $\dbar$-Neumann problem
on $(0,q)$ forms, $0\leq q \leq n$, in the strictly
pseudoconvex non-isotropic 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 |
63. CJM 1997 (vol 49 pp. 798)
Boundedness of solutions of parabolic equations with anisotropic growth conditions In this paper, we consider the parabolic equation
with anisotropic growth conditions, and obtain some criteria on
boundedness of solutions, which generalize the corresponding results
for the isotropic case.
Keywords:Parabolic equation, anisotropic growth conditions, generalized, solution, boundness Categories:35K57, 35K99. |
64. CJM 1997 (vol 49 pp. 232)
Spectral theory for the Neumann Laplacian on planar domains with horn-like ends The spectral theory for the Neumann Laplacian on planar domains with
symmetric, horn-like 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 |