Grant Keady

A semilinear elliptic eigenvalue problem, II. The plasma problem

This paper continues the study of the boundary value problem, for (λ, ψ) Here δ denotes the Laplacian, k is a given positive constant, and λ 1 will denote the first eigenvalue for the Dirichlet problem for −δ on Ω. For λ ≦ λ 1 , the only solutions are those with ψ = 0. Throughout we will be interested in solutions (λ, ψ) with λ > λ 1 and with ψ > 0 in Ω. In the special case Ω = B (0, R ) there is a branch ℱ e , of explicit exact solutions which bifurcate from infinity at λ = λ 1 and for which the following conclusions are valid, (a) The set A ψ , is simply-connected, (b) Along ℱ e , ψ m → k , ‖ψ‖ 1 → 0 and the diameter of A ψ tends to zero as λ → ∞, where Here it is shown that the above conclusions hold for other choices of Ω, and in particular, for Ω = (− a, a )×(− b, b ). (Existence is settled in Part I, and elsewhere.) The results of numerical and asymptotic calculations when Ω = (− a, a )×(− b, b ) are given to illustrate both the above, and some limitations in the conclusions of our analysis.

THE POWER CONCAVITY OF SOLUTIONS OF SOME SEMILINEAR ELLIPTIC BOUNDARY-VALUE PROBLEMS

Soit Ω un domaine convexe borne dans R 2 a frontiere lisse. Soit o<γ<1. Soit u∈C 2 (Ω)∩C(Ω) une solution positive dans Ω, de −Δu=u γ dans Ω, u=o sur ∂Ω. Alors la fonction u α est concave pour α=(1-γ)/2

On functional equations leading to exact solutions for standing internal waves

Abstract The Dirichlet problem for the wave equation is a classical example of a problem which is ill-posed. Nevertheless, it has been used to model internal waves oscillating harmonically in time, in various situations, standing internal waves amongst them. We consider internal waves in two-dimensional domains bounded above by the plane z = 0 and below by z = − d ( x ) for depth functions d . This paper draws attention to the Abel and Schroder functional equations which arise in this problem and use them as a convenient way of organising analytical solutions. Exact internal wave solutions are constructed for a selected number of simple depth functions d .

MapleNet and Maplets under Maple 8

Maplets were reviewed as a package of Maple 7 by Keady [1]. As part of a wider review of Maple 8, McCabe [2] also discusses Maplets. McCabe’s review, as with Keady’s [1] on Maplets in Maple 7, differs from this review in that our focus is on aspects related to MapleNet. (We also happen to think that McCabe may have been overcritical of the amount of detail needed to code Maplets. However, like him, we can also imagine future provision of higher level tools, or Maplet authoring language elements, which make the Maplet author’s coding task less detailed.)