Variational and Topological Methods: Theory, Applications,
Numerical Simulations, and Open Problems.
Electron. J. Diff. Eqns., Conference 21 (2014), pp. 11-21
Title: A model problem for ultrafunctions
Authors: Vieri Benci (Univ. degli Studi di Pisa, Italy)
Lorenzo Luperi Baglini (Univ. of Vienna, Austria)
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.