Two Problems in Potential Theory

It is shown that a class of axially symmetric boundary value problems in potential theory for the region external to a lens may be reduced to problems in two-dimensional potential theory. The solution for the general axially symmetric Dirichlet problem is obtained and it is shown that the Stokes stream function for an arbitrary flow may also be reduced to a Dirichlet problem in potential theory. The Green's function for the lens is also derived and the extension of the approach to non-axially symmetric problems is considered.

Introduction.- Mathematical Preliminaries.- Regularization of moment problems by trancated expansion and by the Tikhonov method.- Backus-Gilbert regularization of a moment problem.- The Hausdorff moment problem: regularization and error estimates.- Analytic functions: reconstruction and Sinc approximations.- Regularization of some inverse problems in potential theory.- Regularization of some inverse problems in heat conduction.- Epilogue.- References.- Index.

This paper gives an integral equation, the solution of which is a solution of a classical problem in potential theory: Given a region with boundary B \mathcal {B} , what distribution of charge on B \mathcal {B} will produce a potential having specified values on B \mathcal {B} ? The paper also indicates briefly how the integral equation is useful in simplifying certain proofs and extending certain theorems in potential theory.

The boundary node method (BNM) is a boundary-only meshfree method based on boundary integral equations (BIE). One drawback of the usual BNM, however, is that it typically requires much more computer time than the usual boundary element method (BEM). The multipole method (MM) has been demonstrated, in the context of the n body problem, and the BEM, to greatly accelerate these methods while still maintaining sufficient accuracy. The present paper explores, for the first time, a coupling of the BNM with the MM (called the BNMM) in the context of 2-D potential theory. Numerical results (for selected problems) from the BNM are compared with those from the BNMM with regard to accuracy and computational efficiency.

Kelvin's transformation is a non-linear map that, in some sense, preserves harmonicity. This property, which was the content of a letter sent by Kelvin to Liouville in 1845, provides a powerful machinery for solving particular potential problems in a very effective way. In the present work, we show that the basic theory can be extended to the biharmonic equation as well to the equations for irrotational and rotational Stokes flow. Hence, biharmonicity, stream functions and bistream functions are also preserved, in some sense, under the Kelvin transformation. We also demonstrate how the Kelvin-type theorems are interconnected with the relative Almansi-type decompositions. These results provide a way to solve analytically many problems in potential theory and Stokes flow which it is impossible to solve by the classical spectral method.

Existence of multipliers for semi-infinite nonlinear programs is established by means of an independence condition of the constraints which stemmed from the study of the conditional Gauss variational problem in potential theory. In the case where the objective function is a quadratic form, our results are generalizations of those for the conditional Gauss variational problem.KeywordsExistence TheoremUnique PointFull SystemIndependence ConditionCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On two free boundary problems in potential theory

Existence of solutions of a nonlinear problem in potential theory.

Application of Bethe-Salpeter methods to some renormalization problems in potential theory: R. Rockmore, Department of Physics, Rutgers University, New Brunswick, New Jersey

The explicit solution of some axialsymmetric scalar screen problems in Sobolev spaces is presented. The well-posedness of the boundary integral equations, which are formulated as dual integral equations, is proved. A reduction for the mixed boundary value problem to a system of singular integral equations with a symbol from the Wiener-algebra is given.The approach of dual integral equations has a long history [Mixed Boundary Value Problems in Potential Theory, North-Holland, Amsterdam, 1966] whereas the new developments reviewed in [E. Meister and F. O. Speck, Modern Wiener–Hopf methods in diffraction theory, Proc. Conf. Dundee, 1988, in Ordinary and Partial Differential Equations, B. Sleeman and R. Jarvis, eds., 1989, pp. 130–171] make it possible to handle these equations in Sobolev spaces.

Potential theory characteristically leads to partial differential equations under boundary conditions. It is shown that these boundary-value problems can be converted directly into equivalent initial-value problems. These initial-value problems are shown to lead to a number of standard and nonstandard numerical procedures.

The distribution of the zeros of the Hermite-Padé polynomials of the first kind for a pair of functions with an arbitrary even number of common branch points lying on the real axis is investigated under the assumption that this pair of functions forms a generalized complex Nikishin system. It is proved (Theorem ) that the zeros have a limiting distribution, which coincides with the equilibrium measure of a certain compact set having the -property in a harmonic external field. The existence problem for -compact sets is solved in Theorem .The main idea of the proof of Theorem consists in replacing a vector equilibrium problem in potential theory by a scalar problem with an external field and then using the general Gonchar-Rakhmanov method, which was worked out in the solution of the ‘’-conjecture.The relation of the result obtained here to some results and conjectures due to Nuttall is discussed.Bibliography: 51 titles.

An examination of two sets of examples shows that the external field arising from the gap in a magnetic digit recording head is very insensitive to the pole-piece shape. It follows that head design should be decided by facility of manufacture rather than by theoretical perfection.

Chapter Three - The BEM for Potential Problems in Two Dimensions

THE need for a treatise on harmonic analysis, more particularly on spherical harmonics, has been recognised for a long time by mathematicians who teach the subject or who find it necessary to use spherical harmonics in their investigations. Those who had the good fortune to be introduced to the study of the subject by the author of the present work had always hoped that he would find time to undertake the preparation and publication of such a treatise. It may be said at once that the work now published fulfils the expectations they had formed. Although many books which contain information about spherical harmonics have appeared since the publication of Heine's “Kugelfune-tionen”, none of them has claims to be considered a complete treatise on the subject, and Heine's work, though it has proved of great service to students of the subject and to mathematicians who had occasion to use harmonic analysis in their investigations, is a compilation of results which contains the material for a treatise rather than a treatise. Much has been written on the subject since the appearance of Heine's two volumes. The mere perusal of the literature involves much labour, and the task of selecting from this mass of literature what is of prime importance to students of the subject and what is necessary for the mathematical physicist is one which demands a wide knowledge of the related branches of mathematics and of applied mathematics. That the author has fulfilled these requirements the present treatise is the evidence. The first four chapters give a complete account of the origin of the ordinary spherical harmonics; Legendre's coefficients of the first and second kinds and the associated functions are treated in detail, their more important properties are developed, and their expression in the form of series and of definite integrals is investigated. Solid harmonics are discussed fully, the addition theorem for ordinary spherical harmonics is established, and the connexion of the functions with potential theory is dealt with. The results given in this part of the treatise are sufficient for the solution of the problems of potential theory and of other branches of mathematical physics where the space under consideration is the space inside or outside the surface of a sphere or the space between two concentric spheres.