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Search: All articles in the CJM digital archive with keyword Laplacian

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1. CJM Online first

He, Jianxun; Xiao, Jinsen
Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two
Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$.

Keywords:Radon transform, wavelet transform, free nilpotent Lie group, unitary representation, inversion formula, sub-Laplacian
Categories:43A85, 44A12, 52A38

2. CJM 2011 (vol 63 pp. 648)

Ngai, Sze-Man
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps
We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities
Categories:28A80, , , , 35P20, 35J05, 43A05, 47A75

3. CJM 2010 (vol 62 pp. 808)

Legendre, Eveline
Extrema of Low Eigenvalues of the Dirichlet-Neumann Laplacian on a Disk
We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet--Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact $1$-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

Keywords: Laplacian, eigenvalues, Dirichlet-Neumann mixed boundary condition, Zaremba's problem
Categories:35J25, 35P15

4. CJM 2010 (vol 62 pp. 1116)

Jin, Yongyang; Zhang, Genkai
Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups
Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \operatorname{div}_X (|\nabla_{X} u|^{p-2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.

Keywords:fundamental solutions, degenerate Laplacians, Hardy inequality, H-type groups
Categories:35H30, 26D10, 22E25

5. CJM 2006 (vol 58 pp. 449)

Agarwal, Ravi P.; Cao, Daomin; Lü, Haishen; O'Regan, Donal
Existence and Multiplicity of Positive Solutions for Singular Semipositone $p$-Laplacian Equations
Positive solutions are obtained for the boundary value problem \[\begin{cases} -( | u'| ^{p-2}u')' =\lambda f( t,u),\;t\in ( 0,1) ,p>1\\ u( 0) =u(1) =0. \end{cases} \] Here $f(t,u) \geq -M,$ ($M$ is a positive constant) for $(t,u) \in [0\mathinner{,}1] \times (0,\infty )$. We will show the existence of two positive solutions by using degree theory together with the upper-lower solution method.

Keywords:one dimensional $p$-Laplacian, positive solution, degree theory, upper and lower solution
Category:34B15

6. CJM 2006 (vol 58 pp. 381)

Jakobson, Dmitry; Nadirashvili, Nikolai; Polterovich, Iosif
Extremal Metric for the First Eigenvalue on a Klein Bottle
The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a sphere ${\mathbb S}^4$ by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawson's $\tau_{3,1}$-torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for $\lambda_1$ on a Klein bottle of a given area. We present numerical evidence and prove the first results towards this conjecture.

Keywords:Laplacian, eigenvalue, Klein bottle
Categories:58J50, 53C42

7. CJM 2006 (vol 58 pp. 64)

Filippakis, Michael; Gasiński, Leszek; Papageorgiou, Nikolaos S.
Multiplicity Results for Nonlinear Neumann Problems
In this paper we study nonlinear elliptic problems of Neumann type driven by the $p$-Laplac\-ian differential operator. We look for situations guaranteeing the existence of multiple solutions. First we study problems which are strongly resonant at infinity at the first (zero) eigenvalue. We prove five multiplicity results, four for problems with nonsmooth potential and one for problems with a $C^1$-potential. In the last part, for nonsmooth problems in which the potential eventually exhibits a strict super-$p$-growth under a symmetry condition, we prove the existence of infinitely many pairs of nontrivial solutions. Our approach is variational based on the critical point theory for nonsmooth functionals. Also we present some results concerning the first two elements of the spectrum of the negative $p$-Laplacian with Neumann boundary condition.

Keywords:Nonsmooth critical point theory, locally Lipschitz function,, Clarke subdifferential, Neumann problem, strong resonance,, second deformation theorem, nonsmooth symmetric mountain pass theorem,, $p$-Laplacian
Categories:35J20, 35J60, 35J85

8. CJM 2005 (vol 57 pp. 1279)

Maad, Sara
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

9. CJM 2000 (vol 52 pp. 1057)

Urakawa, Hajime
The Spectrum of an Infinite Graph
In this paper, we consider the (essential) spectrum of the discrete Laplacian of an infinite graph. We introduce a new quantity for an infinite graph, in terms of which we give new lower bound estimates of the (essential) spectrum and give also upper bound estimates when the infinite graph is bipartite. We give sharp estimates of the (essential) spectrum for several examples of infinite graphs.

Keywords:infinite graph, discrete Laplacian, spectrum, essential spectrum
Categories:05C50, 58G25

10. 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 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

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