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$$(\mathcal C,\mathcal E,\mathcal P)$$-sheaf structures and applications. (English) Zbl 0938.35008
Grosser, Michael (ed.) et al., Nonlinear theory of generalized functions. Proceedings of the workshop on nonlinear theory of nonlinear functions, Erwin-Schrödinger-Institute, Vienna, Austria, October-December 1997. Boca Raton, FL: Chapman & Hall. Chapman Hall/CRC Res. Notes Math. 401, 175-186 (1999).
The author constructs a sheaf $${\mathcal A}$$ of $$({\mathcal C},{\mathcal E},{\mathcal P})$$-algebras, where $${\mathcal C}$$ denotes a ring of generalized numbers, $${\mathcal E}$$ a sheaf of algebras on a topological space $$X$$ and, with $$\Omega$$ ranging in the open subsets of $$X,$$ $${\mathcal P} (\Omega)$$ a family of seminorms on $${\mathcal E}(\Omega).$$ $${\mathcal A}$$ generalizes many previous approaches in the theory of generalized functions. The sheaf structure allows one to define localization and microlocalization. The author then gives applications to a perturbation problem of the kind $\varphi(\varepsilon) dX/dt=f_\varepsilon(t,X),\quad X(0)= \psi(\varepsilon),$ upon suitably interpreting the problem as a fixed point problem in some algebras whose parameters $$({\mathcal C},{\mathcal E},{\mathcal P})$$ are adjusted to the perturbation.
For the entire collection see [Zbl 0918.00026].

MSC:
 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs 32C38 Sheaves of differential operators and their modules, $$D$$-modules
Keywords:
generalized functions