Differential calculus and integration of generalized functions over membranes

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13–29, 2005 ). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion o...

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Bibliographic Details
Published in:Monatshefte für Mathematik Vol. 166; no. 1; pp. 1 - 18
Main Authors: Aragona, Jorge, Fernandez, Roseli, Juriaans, Stanley O., Oberguggenberger, Michael
Format: Journal Article
Language:English
Published: Vienna Springer Vienna 01.04.2012
Subjects:
Mathematics
Mathematics and Statistics
Membranes
26E30
Generalized functions
30G06
46T20
46F30
Non-Archimedean differential calculus
Colombeau algebras
Generalized Cauchy formula
ISSN:0026-9255, 1436-5081
Online Access:Get full text
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Summary:In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13–29, 2005 ). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144:13–29, 2005 ), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green’s theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-010-0275-z