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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Vydáno v:	Monatshefte für Mathematik Ročník 166; číslo 1; s. 1 - 18
Hlavní autoři:	Aragona, Jorge, Fernandez, Roseli, Juriaans, Stanley O., Oberguggenberger, Michael
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Vienna Springer Vienna 01.04.2012
Témata:	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
On-line přístup:	Získat plný text
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Popis
Shrnutí:	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