2012/12/06 by Vieri Benci, Benci, Vieri, Lorenzo Luperi Baglini +1
Mathematics · #26E30 #26E35 #35D99 #35J57 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:26E30 #msc:26E35 #msc:35D99 #msc:35J57
paper · pdf · doi:10.48550/arxiv.1212.1370
14 pages
arxiv created 2012/12/06 · arxiv updated 2012/12/07
The purpose of this paper is to 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∫Omega|∇ u(x)|2dx+u(q). If Omega is a subset of RN is a bounded open set and u is a C2 function with condition u=0 on the boundary of Omega, this problem has no solution since inf E(u,q)=-∞. On the contrary, as we will show, this problem is well posed in a suitable non-Archimedean frame. More precisely, we apply thegeneral 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.