If Ω is a bounded domain in Rn satisfying certain conditions, Ωk denotes its intersection with a k-dimensional hyperplane, 1 ≦ k ≦ n, it is shown that the embedding of the Sobolev space Ws,p(Ω), s>0, into Lq(Ωk) is of type lm if for q<p<∞. The same result is obtained for the space of Bessel potentials Ls,p(Ω). Piecewise polynomial and Fourier approximations of functions and interpolation theorems areused.

The positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.

In answer to two questions raised by W. N. Everitt, we show that, given p > l and any countably infinite set of isolated points on the positive λ-axis, there is a q(x) in Lp(0, ∞) for which the set of points constitutes the point-continuous spectrum associated with the equation y”(x) + {λ − q(x)}y(x) = 0 (0≦x<∞) and some homogeneous boundary condition at x = 0.

This paper provides a survey on a class of methods to obtain sufficient conditions for the inversemonotonicity of second-order differential operators. Pointwise differential inequalities as well as weak differential inequalities are treated. In particular, the theory yields results on the relation between inverse-mo no tone operators and monotone definite operators, i.e. monotone operators in the Browder–Minty sense. This presentation is restricted to ordinary differential operators. Most methods explained here can also be applied to elliptic-parabolic partial differential operators in essentially the same way.

We classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid = 1, finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.

The existence of zeroes of an operator in Banach space near a. known zero is examined, when the non-degeneracy condition of Magnus [1] is violated, and replaced by a condition which is called simple-degeneracy. Criteria are given which help determine the number and the structure of curves of zeroes of the operator, passing through the known zero.

These conditions and results are interpreted in terms of bifurcation theory. By considering several different cases, it is shown that the failure of various conditions frequently employed in the Lyapunov–Schmidt method may be overcome using the idea of simple-degeneracy.

A new method is developed for identifying real-valued coefficients r(x), p(x), and q(x) for which all solutions of the fourth-order differential equation

are L2(0, ∞). The results are compared with those derived from the asymptotic theory of Devinatz, Walker, Kogan and Rofe-Beketov.

The symmetric differential expression M determined by Mf = − Δf;+qf on G, where Δ is the Laplacian operator and G a region of n-dimensional real euclidean space Rn, is said to be separated if qfϵL2(G) for all f ϵ Dt,; here D1 ⊂ L2(G) is the maximal domain of definition of M determined in the sense of generalized derivatives. Conditions are given on the coefficient q to obtain separation and certain associated integral inequalities.

Let T be the formally self-adjoint second-order elliptic differential expression

in ℝn, where the coefficients bj, ajk, q are real-valued and ajk ≈ akj. In this paper sufficient conditions for all positive integer powers Tm of T to be essentially self-adjoint on are obtained.

We develop various facets of the theory of quadratic forms on a Hilbert space suggested by a criterion of Kato which characterizes closed forms in terms of lower semicontinuity.

When a sheet of paper is crumpled in the hands and then crushed flat against a desk-top, the pattern of creases so formed is governed by certain simple rules. These rules generalize to theorems on folding Riemannian manifolds isometrically into one another. The most interesting results apply to the case in which domain and codomain have the same dimension. The main technique of proof combines the notion of volume with Hopf's concept of the degree of a map.

The perturbed linear ordinary differential equation

is considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

The groups of the title are characterized arithmetically.

A scheme devised by Chandrasekhar for investigating the transformations between various differential equations of the second order governing perturbations of the Schwarzschild black hole demands further investigation. The transformation between two differential equations in normal form is considered, and a wide survey of the properties of the transformation is given. It is shown how Chandrasekhar's equations fit into the scheme, after which some examples with particular properties are considered. A detailed investigation of Bessel's equation is undertaken using various devices, in particular by employing asymptotic methods for products of Bessel functions, and employing matrix methods for dealing with large numbers of matrix equations which necessitates an interesting method of solution, the results being reinterpretations of the standard recurrence relations for Bessel functions.

Using Hilbert space methods, existence and uniqueness are proved for the solution of some strongly non-linear partial differential equations of elliptic and parabolic type.

They are associated with quasi-linear operators of the form: -div(β(x, grad u)) + β0(x, u) where β (resp β0) is a maximal monotone subdifferential on ℝN(resp ℝ) depending smoothly on x in a bounded domain Ω of ℝN

These operators are shown to be the subdifierentials over Lp(Ω) of convex functional of the following type:

where j is a normal convex integrand over Ω×ℝN+1 satisfying a coerciveness condition.

This method avoids the theory of Sobolev-Orlicz spaces. An application is given also forthe gas-diffusion equation over ℝ+.

Formulae are derived for the numbers of completely 0-simple and completely simple semigroups with m ℒ-classes, n ℜ-classes and underlying finite group G.

It is shown that the fine structure of the asymptotic estimates for the eigenvalues of a large class of ordinary differential operators, can be described in terms of the Fourier coefficients of a function of class L2.

The operator equation TS = 1 is studied for power bounded operators T, S on Hilbert space, and its relation to *—representations of the bicyclic semigroup is explored.

Given the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.

A unified description and treatment is given for a large and important class of homogeneous Siegel domains in finite and infinite dimensions. These domains are shown to be linearly equivalent to generalized upper half-planes in spaces of operators having a kind of triple product structure.