W. N. Everitt

PRODUCTS OF DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract To form products of differential expressions in the classical way it is necessary to place heavy differentiability assumptions on the coefficients. Here we consider symmetric (formally self-adjoint) expressions defined, not in the classical way, but in terms of quasi-derivatives. With this very general notion of symmetry we show that products such as M1M2MI of symmetric expressions M1, Hp can be formed vithout any smoothness assumptions on the coefficients and such products are symmetric expressions.

A GENERAL INTEGRAL INEQUALITY ASSOCIATED WITH CERTAIN ORDINARY DIFFERENTIAL OPERATORS

1. This note is concerned with inequalities of the form (1.1) where the open interval (a,b) of integration may be bounded or unbounded, i.e. −∞ ⋚ a ⋚ b ⋚ ∞, the coefficients p, q and w are real-valued on (a,b), w is non-negative, M is the symmetric differential expression (1.2) and η is the largest linear manifold of real-valued functions on (a,b), so chosen that the integrals on the right-hand side are both finite.

On some eigenvalue problems in fuel–cell dynamics

An eigenvalue problem arising from the study of gas dynamics in a loaded tubular solid oxide fuel cell is considered. An asymptotic theory from which the equations are derived is reviewed, and the results of analysis on the small and large parameter asymptotics are presented. These results suggest an interesting and hitherto unknown property of a class of Sturm–Liouville problems in which the first eigenvalue approaches zero but subsequent ones approach infinity as a parameter approaches zero. This was first discovered numerically and later confirmed asymptotically and rigorously.

A NOTE ON AN INTEGRAL INEQUALITY

1. This note is concerned with integral inequalities of the following form (1.1) where α is a real number, f a real-valued function so chosen that the two integrals on the right-hand side are well-defined and finite, and denotes differentiation. The best possible value of K(α) is given for all a and the cases −∞ < α ⋚ −1 are discussed in detail.