Onp-adic Green's functions

Some classes of pseudodifferential operators over the field ofp-adic numbers and local fields with discrete valuation are studied. Sufficient conditions for the Green's functions to be non-negative are found.Institute of Mathematics, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 123–139, July, 1993.     

Green's functions for polynomials in the Laplacian

A Green's function is constructed for an arbitrary polynomial in theN-dimensional Laplacian operator, subject only to the condition that no root of the polynomial may be real and negative.     

On a representation of the Green's functions for shallow panels supported on the boundary of a rectangular base

To enhance the convergence of double series in the case of a shallow panel supported on a rectangular base and a long closed cylindrical shell we propose an approximate representation of the Green's function as a combination of rapidly convergent expansions and a closed-form analytic component obtained by approximate summation of one particular part of series using the two-dimensional integral Fourier transform and reduction to the Kelvin functions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 78–83.     

Finely continuously differentiable functions

This paper establishes an explicit characterization of those real-valued functions on a finely open set in Euclidean space which are continuously differentiable with respect to the fine topology of classical potential theory. It improves and extends previous work of several authors. Differentiability of all orders is considered, and some consequences of the characterization are deduced.     

Finely continuously differentiable functions

In Lávi?ka [A remark on fine differentiability, Adv. Appl. Clifford Algebras 17 (2007) 549–554], it is observed that finely continuously differentiable functions on finely open subsets of the plane are just functions which are finely locally extendable to usual continuously differentiable functions on the whole plane. In this note, it is proved that, under a mild additional assumption, this result remains true even in higher dimensions. Here the word “fine” refers to the fine topology of classical potential theory.     

An initial-value method for the determination of Green's functions

The convergence of an initial-value method for computing the Green's function of a class of second-order differential operators is established. The proof relies on an interpolation procedure which is shown to generalize the Nyström method for Fredholm integral equations. The approximate Green's function is related to the solution of a discrete summation equation. The results of Anselone and Moore on collectively compact operators are then applied.This research was partially supported by UNLV Grant No. 001-060-4573.     

Green's functions for Laplace's equation in multiply connected domains

Analytical formulae for the first-type Green's function forLaplace's equation in multiply connected circular domains arepresented. The method is constructive and relies on the useof a special function known as the Schottky–Klein primefunction associated with multiply connected circular domains.It is shown that all the important functions of potential theory,including the first-type Green's function, the modified Green'sfunctions and the harmonic measures of a domain, can be writtenin terms of this prime function. A broad range of representativeexamples are given to demonstrate the efficacy of the methodas well as a quantitative comparison, in special cases, withthe results obtained using other independent methods.