Space-Time Foam Differential Algebras of Generalized Functions and a Global Cauchy-Kovalevskaia Theorem
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Space-Time Foam Differential Algebras of Generalized Functions and a Global Cauchy-Kovalevskaia Theorem Elemér E. Rosinger
Received: 5 February 2008 / Accepted: 23 September 2008 / Published online: 5 October 2008 © Springer Science+Business Media B.V. 2008
Abstract The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from the so called space-time foam structures in General Relativity, where the set of singularities can be dense. A variety of applications of these algebras have been presented elsewhere, including in de Rham cohomology, Abstract Differential Geometry, Quantum Gravity, etc. Here a global Cauchy-Kovalevskaia theorem is presented for arbitrary analytic nonlinear systems of PDEs. The respective global generalized solutions are analytic on the whole of the domain of the equations considered, except for singularity sets which are closed and nowhere dense, and which upon convenience can be chosen to have zero Lebesgue measure. In view of the severe limitations due to the polynomial type growth conditions in the definition of Colombeau algebras, the class of singularities such algebras can deal with is considerably limited. Consequently, in such algebras one cannot even formulate, let alone obtain, the global version of the Cauchy-Kovalevskaia theorem presented in this paper. Keywords Differential algebras · Generalized functions · Dense singularities · Global Cauchy-Kovalevskaia theorem · Improved smoothness Mathematics Subject Classification (2000) 35xx · 35Axx · 35Dxx · 35Gxx · 46F30
We do not possess any method at all to derive systematically solutions that are free of singularities. . . Albert Einstein The Meaning of Relativity Princeton Univ. Press, 1956, p. 165
E.E. Rosinger () Department of Mathematics and Applied Mathematics, University of Pretoria, 0002 Pretoria, South Africa e-mail: [email protected]
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E.E. Rosinger
1 Algebras of Generalized Functions with Dense Singularities, or Space-Time Foam Algebras 1.1 Families of Dense Singularities in Euclidean Spaces In this paper, following Rosinger [44, 45, 47, 48, 50], we consider differential algebras of generalized functions—called space-time foam algebras—which have significantly strengthened and extended properties with respect to the singularities they can deal with. Namely, this time the singularities can be arbitrary, including dense sets, and the only condition they have to satisfy is that their complementaries, that is, the set of nonsingular points, be also dense. This, among others, allows for singularity sets with a cardinal larger than that of the set of nonsingular points. For instance, in the case the domain is the real line, the set of singularities can be given by the uncountable set of all the irrational numbers, since its complement
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