Non-Standard Asymptotic Analysis and Non-Linear Theory of Generalized Functions

The main purpose of this project is to construct a particular differential algebra ρE(Ω) of generalized functions which, among other things, satisfies: (a) ρE(Ω) is an algebra of Colombeau type, i.e. ρE(Ω) contains a copy of the space D′(Ω) of Schwartz distributions (Schwartz generalized functions on an open set Ω ⊆ Rd). (b) The set of the scalars ρC of ρE(Ω) (ρC consists of the functions in ρE(Rd) with zero gradient) is a Cantor complete algebraically closed field extension of C. The latter is an essential improvement of the original J.F. Colombeau theory. The applications of the algebra ρE(Ω) include: solving linear partial differential equations with variable coefficients or with discontinuous coefficients, solving non-linear partial differential equations originating in mathematical physics and general relativity theory with emphasis on the shock-wave solutions. The research will lead to a series of joint articles published in refereed mathematics journals and a research monograph under agreement with Kluwer Academic Publishers. The participants of this project include several mathematicians in Austria (University of Vienna and University of Insbruck) and a Cal Poly graduate student, Guy Burger. The most essential part of the research will be done at the University of Vienna, Austria, in the period January-July, 2006, but preliminary work is going on now at Cal Poly. TheMathematics Subject Classification: 46F30, 46F10, 46S20, 35D05, 35D10, 35R05, 35L67.