Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems.
Electron. J. Diff. Eqns., Conference 21 (2014), pp. 11-21

A model problem for ultrafunctions

Vieri Benci, Lorenzo Luperi Baglini

Abstract:
In this article. we show that non-Archimedean mathematics (NAM), namely mathematics which uses infinite and infinitesimal numbers, is useful to model some physical problems which cannot be described by the usual mathematics. The problem which we will consider here is the minimization of the functional
$$ E(u,q)=\frac{1}{2}\int_{\Omega }|\nabla u(x)|^2dx+u(q). $$
When $\Omega \subset \mathbb{R}^{N}$ is a bounded open set and $u\in \mathcal{C}_0^2(\overline{\Omega })$, this problem has no solution since $\inf E(u,q)=-\infty$. On the contrary, as we will show, this problem is well posed in a suitable non-Archimedean frame. More precisely, we apply the general ideas of NAM and some of the techniques of Non Standard Analysis to a new notion of generalized functions, called ultrafunctions, which are a particular class of functions based on a Non-Archimedean field. In this class of functions, the above problem is well posed and it has a solution.

Published February 10, 2014.
Math Subject Classifications: 26E30, 26E35, 35D99, 35J57.
Key Words: Non Archimedean mathematics; non standard analysis; ultrafunctions; delta function; Dirichlet problem.

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Vieri Benci
Dipartimento di Matematica
Università degli Studi di Pisa
Via F. Buonarroti 1/c, Pisa, Italy
email: benci@dma.unipi.it
Lorenzo Luperi Baglini
University of Vienna, Faculty of Mathematics
Oskar-Morgenstern-Platz 1
1090 Vienna, Austria
email: lorenzo.luperi.baglini@univie.ac.at

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