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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.