On the foundations of nonlinear generalized functions I

Abstract

We construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces, previous attempts in this direction are unified and completed. Several classification results are achieved and applications to nonlinear differential equations involving singularities are given.

60 pages, LaTeX

References in corpus (1)

• Colombeau's Generalized Functions on Arbitrary Manifolds

Cited by in corpus (31)

• The use of Generalised Functions and Distributions in General Relativity
• Remarks on the distributional Schwarzschild geometry
• Generalised Connections and Curvature
• A global theory of algebras of generalized functions
• Foundations of a nonlinear distributional geometry
• Generalized pseudo-Riemannian geometry
• Aspects of General Relativity: Pseudo-Finsler extensions, Quasi-normal frequencies and Multiplication of tensorial distributions
• Topological structures in Colombeau algebras: topological $\widetilde{\C}$-modules and duality theory
• Spacetimes with distributional semi-Riemannian metrics and their curvature
• Suppleness of the sheaf of algebras of generalized functions on manifolds
• Nonlinear generalized sections and vector bundle homomorphisms in Colombeau spaces of generalized functions
• Symmetry group analysis of weak solutions
• Distributional Schwarzschild Geometry from nonsmooth regularization via Horizon. Disributional Rindler spacetime with disributional Levi-Cività connection induced vacuum dominance.Unruh effect revisited
• A geometric approach to full Colombeau algebras
• Point value characterizations and related results in the full Colombeau algebras G^e and G^d
• Generalized Group Actions in a Global Setting
• On a nonlinear Peetre's theorem in full Colombeau algebras
• Diffeomorphism invariant Colombeau Algebras. Part I: Local theory
• Nonlinear distributional geometry and general relativity
• Diffeomorphism invariant Colombeau algebras. Part III: Global theory
• Sheaves of nonlinear generalized function spaces
• Full and special Colombeau algebras
• Nonlinear generalised functions on manifolds
• Non-smooth differential geometry and algebras of generalized functions
• Approximation properties of local smoothing kernels
• Dirac Processes and Default Risk
• Sheaves of nonlinear generalized functions and manifold-valued distributions
• Intrinsic characterization of manifold-valued generalized functions
• Generalized functions valued in a smooth manifold
• Generalized flows and singular ODEs on differentiable manifolds
• Diffeomorphism invariant Colombeau algebras. Part II: Classification