Expand all Collapse all | Results 1 - 2 of 2 |
1. 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 |
2. 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 |