Fractional integrals and weakly singular integral equations of the first kind in the \(n\)-dimensional ball
type: Item
instance of: scholarly article
title: Fractional integrals and weakly singular integral equations of the first kind in the \(n\)-dimensional ball
zbMATH Open document ID: 0828.45003
DOI: 10.1007/BF03008419
published in: Journal d'Analyse Mathématique
publication date: 1996-01-11T00:00:00Z
review text: The paper is devoted to study the solvability of the integral equation of the first kind \[ (I^\alpha_\Omega \varphi) (x) \equiv c_{n, \alpha} \int_\Omega {\varphi (y) dy \over |x - y |^{n - \alpha}} = f(x), \quad x \in \Omega,\;0 < \alpha < n,\;n \geq 1, \tag{1} \] with \(c_{n, \alpha} = 2^{- \alpha} \pi^{- n/2} \Gamma [(n - \alpha)/2]/ \Gamma (\alpha/2)\) in an \(n\)-dimensional ball \(\Omega \equiv U_c = \{t \in R^n : |t |< c\}\) \((0 < c < \infty)\), a disk when \(n = 2\). Such equations arise in applied problems in which it is important to know solvability conditions in special functional spaces and explicit formulae of solutions. The formal solution of equation (1) on a disk was obtained as a series by \textit{E. T. Copson} [Proc. Edinb. Math. Soc. 8, 14-19 (1947)] and \textit{N. A. Rostovtsev} [J. Appl. Math. Mech. 25, 238-246 (1961; Zbl 0107.194)] and via fractional differentiation operators by \textit{V. I. Fabrikant} [Izv. Vyssh. Uchebn. Zaved., Mat. 1971, No. 2(105), 102-104 (1971; Zbl 0221.45006)]. Sufficient conditions for the existence of at most one solution of equation (1) and explicit forms of such solutions were given by \textit{Ch. S. Kahane} [Integral Equations Oper. Theory 4, 548-595 (1981; Zbl 0472.45001)]\ for a disk and by \textit{Ch. S. Kahane} [ibid. 6, 67-133 (1983; Zbl 0516.45001)] and the author [Scientific works of the Jubilee seminar on boundary value questions, Collect Artic., Minsk 1985, 181-185 (1985; Zbl 0623.45013)] for a ball. In this connection we note the papers of \textit{N. K. Karapetjanc} and the author [Sov. Math., Dokl. 25, No. 6, 522-525 (1982; Zbl 0566.45020)], the author [Math. Notes 34, 751-757 (1983; Zbl 0535.45008)] and the author [Soviet Math. (Iz. VUZ) 29, No. 6, 102-106 (1985; Zbl 0601.31004)] where the invertibility of the Riesz potential \(I^\alpha_{U_c} \varphi\) with the radial \(\varphi (|x |)\)- and general \(\varphi (x)\)- density was investigated. The papers above did not contain the conditions of solvabil…
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 569416
zbMATH Keywords: weakly singular integral equations
zbMATH Keywords: multidimensional integral equations of the first kind with power kernel
zbMATH Keywords: multidimensional fractional integrals
zbMATH Keywords: Riesz potential
zbMATH Keywords: solvability conditions
author: Boris Rubin
reviewed by: A. A. Kilbas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 569416