2012/06/02 by Vieri Benci, Benci, Vieri
Mathematics · #26E30 #26E35 #35D99 #81Q99 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:26E30 #msc:26E35 #msc:35D99 #msc:81Q99
paper · pdf · doi:10.48550/arxiv.1206.2257
34 pages
arxiv created 2012/09/06 · arxiv updated 2012/09/07
The theory of distributions provides generalized solutions for problems which do not have a classical solution. However, there are problems which do not have solutions, not even in the space of distributions. As model problem you may think of -\Deltau=up-1, u>0, p≥(2N)/(N-2) with Dirichlet boundary conditions in a bounded open star-shaped set. Having this problem in mind, we construct a new class of functions called ultrafunctions in which the above problem has a (generalized) solution. In this construction, we apply the general ideas of Non Archimedean Mathematics and some techniques of Non Standard Analysis. Also, some possible applications of ultrafunctions are discussed.