Methods of functional analysis for generalized operators

Over the past decade the theory of Colombeau algebras of generalized functions has answered a wealth of questions of analytic and geometric nature and in mathematical general relativity. Particular attention has been devoted to linear and nonlinear partial differential equations involving non-smooth coefficients and strongly singular data, whose solvability cannot be investigated by means of classical methods. On the other hand, a topological theory of Colombeau algebras, based on an appropriate notion of generalized topological modules, and the corresponding duality theory have been elaborated recently. This project will focus on the question of existence and qualitative properties of solutions to partial differential equations in the Colombeau setting. Functional analytic tools will be developed that incorporate the non-classical spaces of Colombeau theory, notably the topological modules and rings arising there. In particular, this new machinery will provide surjection theorems as well as necessary and sufficient conditions for solvability. In this way, the vast field of linear and nonlinear operator theory will be incorporated in the theory of generalized function algebras and new paths will be opened in Colombeau theory. Further, the theory of variational inequalities and monotone operators, which is of central importance in the modelling of linear and nonlinear problems in science and engineering, will be extended by means of the new functional analytic methods to be developed in the Colombeau framework. This way, their range of applicability will be significantly enlarged, rendering variational methods and monotone operator techniques amenable to strong singularities in the data and coefficients. A detailed investigation of the concrete case of generalized Fourier integral operators will be the second important topic of the project. Motivated by applications in the geometric theory of partial differential equations and mathematical geophysics, a theory of generalized pseudo- and Fourier integral operators will be developed on Colombeau algebras on manifolds. A full understanding of the action of Fourier integral operators and propagation of singularities will be obtained by generalized microlocal analysis techniques elaborated on manifolds and on the duals of Colombeau algebras.