\((\mathcal C,\mathcal E,\mathcal P)\)-sheaf structures and applications. (English) Zbl 0938.35008

Marti, Jean-Andre

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].

Reviewer: A.Parmeggiani (Bologna)

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