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On the foundations of nonlinear generalized functions. I, II. (English) Zbl 0985.46026

Mem. Am. Math. Soc. 729, xi, 93 p. (2001).
A diffeomorphism-invariant differential algebra of generalized functions of Colombeau type \({\mathcal G}^d(\Omega)\) is constructed with a canonical copy of the sapce of Schwartz distributions \({\mathcal D}'(\Omega)\). The result is important for the nonlinear theory of generalized functions and for functional analysis in general. It opens the door to applications to differential geometry, partial differential equations with origin in geometry, and relativity theory.

MSC:

46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
26E15 Calculus of functions on infinite-dimensional spaces
46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
35D05 Existence of generalized solutions of PDE (MSC2000)
35L70 Second-order nonlinear hyperbolic equations
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