Distribution of eigenfrequencies for the wave equation in a finite domain. i. three-dimensional problem with smooth boundary surface.

An asymptotic expansion for the heat equation

Expressions are derived for the free energy of a massless scalar gas confined to a spatial cavity in a static space-time at a finite temperature. A high temperature expansion is presented in terms of the Minakshisundaram coefficients. This gives curvature and boundary corrections to the Planckian form. The regularisation used is the zeta function one, and yields a finite total internal energy. However, it is known that the local energy density diverges in a non-integrable way as the boundary is approached. A 'surface energy' is suggested to reconcile these two facts. Explicit expressions for the total energy inside two infinite rectangular waveguides are obtained.

http://iopscience.iop.org/article/10.1088/0305-4470/11/5/020/pdf

Finite temperature and boundary effects in static space-times

The eigenvalue problem for the Laplace operator in two dimensions is classical in mathematics and physics. Nevertheless, computational methods for estimating the eigenvalues are still of much current interest, particularly in applications to acoustic and electromagnetic waveguides. Although our primary interest is with the computational methods, there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and, in addition, are of interest in themselves. These results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds. Some of the more powerful techniques available are illustrated by applying them to a model problem.

/pdf/eigenvalues-of-the-laplacian-in-two-dimensions-21xy0ib3p2.pdf

Eigenvalues of the Laplacian in Two Dimensions

This paper, somewhat delayed in the series “Fundamental Directions,” is devoted to general linear elliptic boundary problems on a smooth compact manifold with boundary. The paper is intended for a large circle of readers. We hope the paper will be useful to many mathematicians with diverse scientific interests. The main features of the theory we will discuss were formed in the 60’s, beginning with the proof of the equivalence of the ellipticity conditions and the Fredholm property of the corresponding operator in the simplest Sobolev spaces. This was done on the basis of achievements of many mathematicians during the preceding decades. This basis was very extensive, but the results were incomplete. The elaboration of the general theory was stimulated by investigations of the index problem (see e.g. (Palais 1965) and (Fedosov 1990)) and went on under a strong influence of the microlocal analysis, beginning with the appearance of the calculus of pseudodifferential operators. During the last decades, new variants of the general theory appeared, and the old variants were enriched by new results.

Elliptic Boundary Problems

Three classes of models of renewable resource economics are discussed in this paper: profit-maximizing models, competitive equilibrium models, and cooperative equilibrium models. A variety of complexities, both biological and economic, are also discussed, including: irreversibility problems, disaggregated biological models, delay effects, and stochastic models. Questions deserving further research are raised.

Colin W. Clark

Papers

Mathematical Models in the Economics of Renewable Resources

The Asymptotic Distribution of Eigenvalues and Eigenfunctions for Elliptic Boundary Value Problems