The two basic problems solved here are (1) a disc charged to a uniform potential placed between two earthed planes parallel to its plane, each a small distance h from the disc, (2) the space filled with viscous fluid and the disc slowly rotating uniformly about its axis with the planes held at rest. The total charge or turning moment are calculated as series in powers of h, giving four terms in each case. An approximate method devised by Maxwell [5] is shown to have three terms correct in both cases.

The first part of this paper is devoted to a study of the classical bifurcation problem in a Hilbert space, under the assumption that the operators involved are gradient operators, but not necessarily compact. Our approach to the problem was introduced by Krasnosel'skii, but here we show that his assumption about the compactness of the operators can be replaced by a much weaker Lipschitz type condition, without affecting the generality of his conclusions.

The rest of the paper is concerned with the analogous problem when the operator is knownto be asymptotically linear rather than Fréchet differentiable. Indeed, we show that this question can always be reduced to the first case, after some manipulation. After this manipulation the new operator is found to be a Fréchet differentiable gradient operator, and so we can invoke the results of the first part. This manipulation is in the spirit of that of [11] but is necessarily different.

In the Hilbertspace with (,) as inner product we consider the linear operator L with domain D(L) and the sesqui-linear form 〈, 〉 defined by

Let the symmetric operator L0 be the restriction of L to D(L0), where

The propagation of sound from a point source placed in a subsonic jet separated by two plane vortex sheets from two semi-infinite still media is considered and it is found that instability waves arise at particular points on the vortex sheets and that their effect is confined to certain regions.

Two eigenvalue problems associated with steady rotations of a chain are considered. To compare the spectra of these two problems, let σK(n) denote the set of all angular velocities with which a chain of unit length with one end fixed and the other free can rotate in a vertical plane so as to have exactly n nodes on thevertical axis (including the fixed end). In a linearised theory σK(n) is a single point, i.e.

In the full non-linear theory σK(n) is an infinite interval lying to the right of ⍵n. Indeed,

This is established in [1].

Next, let σM(β, n) denote the set of all angular velocities with which a chain, having ends fixed at unit distance apart on the vertical axis, can rotate in a vertical plane so as to have exactly n+l nodes on the vertical axis (including the ends) and so that the tension takes the value β at the lower end. This problem, in which the length of the chain is not prescribed is a model for a spinning process in which the ‘chain’ is continuously created in a rotating configuration. For β>0, we again have in the linearised theory that σM(β, n) is a singleton, i.e.

In the full non-linear theory σM(β, n) lies to the left of λn(β). Although unable to determine exactly σM(β, n) for β>0, we have

where ⍵n λn(β), and are all characterised as the nth zeros of known combinations of Bessel functions.

Though it is still an open problem for which class of first-order elliptic systems Carleman's theorem holds, this is proven here for a certain class of systems (with analytic coefficients) for which Douglis introduced the hypercomplex algebra and hyperanalytic functions. The proof is based on a representation formula generalising Vekua's approach with Volterra integral equations in C2 to more than two unknowns. The representation formula is of its own interest because it provides the generation of complete families of solutions. The equations of plane inhomogeneous elasticity problems lead to a system of the desired class.

This paper considers a natural extension of the following inequality of Hardy and Littlewood

to an inequality of the form

where μ>0 and 0<K(μ)<∞. The best possible value of k(μ) is determined, and all the cases of equality.

The relationship between weak and sequential weak continuity for mappings between Banach spaces and semigroups on Banach space is studied.

Conditions are obtained which ensure that a 2nth order ordinary differential equation on a halfaxis have at most n linearly independent square-integrable solutions. The emphasis is on bounds on the integrals of the coefficients, and on conditions imposed over a sequence of intervals.

The present paper is concerned with the problem of regularity of weak solutions of boundary value problems. We shall present a new method to prove differentiability on the boundary. This method was developed in our thesis [12] within the theory of abstract Sobolev spaces, introduced by Stummel [14]. Here, we shall describe it by applying it to elliptic boundary value problems. It will be seen that the advantage of this method consists in the fact that it is based on functional analysis only and therefore may be used for other types of differential equations as well.

An acoustic line source is moving parallel to the vortex sheet dividing two fluids in relative motion and the lifetime of the source is finite. The resulting phenomena are compared with those created by a fixed pulsed source. It is found that the waves produced by the source in motion are not so singular as, though qualitatively similar to, those due to a source which remains in one position but Helmholtz instability still has a dominant part to play. This statement is true whether the source be moving subsonically or supersonically so long as it has non-zero velocity.

Let u be a real valued strong solution defined in a cylindrical domain of a linear second-order parabolic equation in two space variables with entire coefficients. Then it is shown that on compact subsets of its domain of definition u can be approximated arbitrarily closely in the maximum norm by an entire solution of the parabolic equation.

In this paper we consider the problem of defining a convolution, analogous to the classical convolution of scalar measures, of measures defined on the Borel sets of a locally compact semi-group S and having values in a Banach algebra . Using the bilinearintegral introduced by Bartle we show that the formalism of the scalar case persists in situations of considerable generality so that the formula

suitably interpreted, gives a Banach algebra structure to a large class of valued measures defined on S. Themethods exploit the connection between vector measures and operators and involve some results of independent interest.

In this article we examine the qualitative behaviour of non-planar equilibrium states ofnon-linearly elastic rods subject to terminal loads. In our geometrically exact theory, a rod is endowed with enough geometric structure for it to undergo flexure, torsion, axial extension, and shear. The constitutive equations give appropriate stress resultants and couples as non-linear functions of appropriate strains. These constitutive relations must meet minimal conditions ensuring that they be physically reasonable. It turns out that the equilibrium states of such a rod are governed by a boundary value problem for a quasilinear fifteenth-order system of ordinary differential equations.

In this paper the Weyl limit-point and limit-circle theory of second-order differential equations is extended to the case that the weight function is allowed to take on both positive and negative values—the polar case. This extension is achieved using Weyl's limit circle method.

Integral inequalities are used to obtain comparison theorems for a class of second-order differential equations which includes the Emden-Fowler equation, certain Liénard equations, and linear equations of the form d2y/dx2+f(x)y = 0. For these linear equations the results below imply Sturm's classical comparison theorem.

This paper is concerned with solutions of the ordinary differential equation

where ℒ is a real formally self-adjoint, linear differential expression of order 2n, and the perturbed term f satisfies

for some σ∈[0, 1]. Here λ(·) is locally integrable on [0,∞).

In particular it is shown, under circumstances detailed in the text, that (*) possesses solutions in the Hilbert function space L2(0,∞).