A nonstandard generalization of the space of distributions and its applications to PDEs

Place: Seminar Room "-1" -  Department of Mathematics - Via Sommarive 14 - Povo - Trento
at 16.30 p.m.

• Emanuele Bottazzi - PhD in mathematics

Abstract:
In the mathematical practice, whenever a problem does not allow for a solution, suitable notions of generalized solutions are developed, usually by extending the class of admissible solutions. We introduce an algebra of grid functions of nonstandard analysis defined on a hyperfinite domain, and show that this algebra provides a non-trivial generalization of the space of distributions and of the space of Young Measures. In this algebra, the distributional derivative is represented by a finite difference operator of an infinitesimal step. This operator satisfies all of the properties of the classic derivative up to an infinitesimal perturbation, thus partially overcoming the limitations of the negative result by Schwartz [1]. By working with grid functions, we can formulate a wide range of problems from functional analysis in a way that is coherent with the classic theory. As an example, we discuss the grid function formulation of a class of partial differential equations with variable parabolicity direction. In particular, we show that the solutions obtained by this nonstandard formulation generalize the measure-valued solutions for this class of equations. Thanks to this result, we are able to give a positive answer to a conjecture by Smarrazzo [2] regarding the asymptotic behaviour of the measure-valued solutions. 

[1] L. Schwartz, Sur l’impossibilité de la multiplication des distributions, C. R. Acad. Sci. Paris, 29 (1954), pp. 847–848.
[2] F. Smarrazzo, On a class of equations with variable parabolicity direction, Discrete and continuous dynamical systems 22 (2008), pp. 729–758.

Supervisor: Vieri Benci (Università di Pisa)