A model problem for ultrafunctions

IIn 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) = 1/2 integral(Omega)vertical bar del u(x)vertical bar(2) dx + u(q). When Omega subset of R-N is a bounded open set and u is an element of C-0(2) (Omega), this problem has no solution since inf E (u, q) = -infinity. 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.