Differential calculus and integration of generalized functions over membranes
Differential calculus and integration of generalized functions over membranes
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DOI：
10.1007/s00605-010-0275-z
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发表时间：
2012-04-01
影响因子：
0.9
通讯作者：
Oberguggenberger, Michael
中科院分区：
文献类型：
--
作者：
Aragona, Jorge;Fernandez, Roseli;Oberguggenberger, Michael
在本文中，我们继续发展的微分开始在阿拉戈纳等人。144：13-29，2005）。由所谓的尖锐拓扑和Colombeau广义函数作为广义点集上的点函数的解释的指导下，我们引入膜的概念并扩展了Aragona等人（Monatsh. 144：13-29，2005），到膜上定义的积分。我们用它来证明一个广义版本的柯西公式，并获得广义全纯函数的Goursat定理。将经典微积分中的一些结果，如反函数定理、隐函数定理、绿色定理等，推广到广义情形。此外，我们指出，迁移和波动方程的广义初始数据的解公式，以及可以得到。
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.