Colombeau’s Theory of Generalized Functions

Grosser, Michael; Kunzinger, Michael; Oberguggenberger, Michael; Steinbauer, Roland

doi:10.1007/978-94-015-9845-3_1

Cited by 24 publications

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“…by considering indices of the form ε = (r, ϕ). We finally remark that, similarly to the case of G d , one can treat the algebras G 2 andĜ of [9].…”

Section: Big-o For Uniform Asymptotic Relationsmentioning

confidence: 97%

“…Let ⊆ R n be an open set and (u ε ) ∈ C ∞ ( , R) be a net of smooth functions. We use the notations of [9] for moderate and negligible nets related to the special algebra G s ( ), and the notations of [20] for similar notions related to the algebraĜ( ) of asymptotic functions. Then…”

Section: Corollary 215mentioning

confidence: 99%

“…For a more detailed discussion of these and further applications in general relativity we refer to [13], [19], [21]. Concerning Colombeau algebras in general, we refer to [3]- [5], [9], [14] for detailed information; our terminology and notation mainly follows [9]. Besides Colombeau's original algebra, the full algebra G e and the special algebra G s on open subsets of R n appeared (see [3]- [5]) and some years later the diffeomorphism invariant local algebra G d was constructed in [9].…”

mentioning

confidence: 99%

“…(ii) ∀K ∀α ∈ N n ∀m ∈ N : sup x∈K ∂ α u ε (x) = O (ε m ) as ε ∈ I. P r o o f .The proof is only a reformulation of Thm. 1.2.3 of[9], provided we use ε instead of ε. We only note that in one of the final steps we have∂ i u ε (x) = u ε (x + ε m+N e i ) − u ε (x) big-Oare both of the same type O I if I ∈ I s ,Î, I e .…”

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confidence: 99%

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Unifying order structures for Colombeau algebras

Giordano

Nigsch

2015

Math. Nachr.

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We define a general notion of set of indices which, using concepts from pre‐ordered sets theory, permits to unify the presentation of several Colombeau‐type algebras of nonlinear generalized functions. In every set of indices it is possible to generalize Landau's notion of big‐O such that its usual properties continue to hold. Using this generalized notion of big‐O, these algebras can be formally defined the same way as the special Colombeau algebra. Finally, we examine the scope of this formalism and show its effectiveness by applying it to the proof of the pointwise characterization in Colombeau algebras.

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“…by considering indices of the form ε = (r, ϕ). We finally remark that, similarly to the case of G d , one can treat the algebras G 2 andĜ of [9].…”

Section: Big-o For Uniform Asymptotic Relationsmentioning

confidence: 97%

Section: Corollary 215mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Unifying order structures for Colombeau algebras

Giordano

Nigsch

2015

Math. Nachr.

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“…e.g. the survey in [28]), we can stay in the well-established realm of classical, i.e. linear, distribution theoryà la Schwartz.…”

Section: The Lichnerowicz Conditions For Weak Solutionsmentioning

confidence: 99%

Nonanalytic extensions of the extreme Reissner–Nordström metric in terms of weak solutions

Graf¹

2001

Class. Quantum Grav.

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Abstract.A basic extension of the exterior part of the extreme Reissner-Nordström solution in terms of a continuous metric and gauge potential is constructed. This extension is not smooth at the null hypersurface given by the Cauchy-Killing horizon which separates isometric copies of the exterior metric. The Maxwell-Einstein system of equations is satisfied only in a weak sense. The manifold is topologically incomplete and the spherical symmetry is globally broken down to an axial symmetry. This behaviour can be attributed to the effect of a 'topological string', in the sense of a infinitesimally thin closed stringlike object 'sitting on the rim' of the black hole and holding it open by means of an accompanying impulsive gravitational wave. The resulting differentiable manifold and the corresponding horizons are not anymore simply connected, being 'pierced' by the strings.

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Wave Equations on Non-smooth Space-times

Hörmann

Kunzinger

Steinbauer

2012

Evolution Equations of Hyperbolic and Schrödinger Type

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We consider wave equations on Lorentzian manifolds in case of low regularity. We first extend the classical solution theory to prove global unique solvability of the Cauchy problem for distributional data and right hand side on smooth globally hyperbolic space-times. Then we turn to the case where the metric is non-smooth and present a local as well as a global existence and uniqueness result for a large class of Lorentzian manifolds with a weakly singular, locally bounded metric in Colombeau's algebra of generalized functions.

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Colombeau’s Theory of Generalized Functions

Cited by 24 publications

References 0 publications

Unifying order structures for Colombeau algebras

Unifying order structures for Colombeau algebras

Nonanalytic extensions of the extreme Reissner–Nordström metric in terms of weak solutions

Wave Equations on Non-smooth Space-times

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