Ultrafunctions and generalized solutions

TL;DR

This paper introduces ultrafunctions, a new class of generalized functions based on Non-Archimedean Mathematics, enabling solutions to problems without classical or distributional solutions, with potential applications in nonlinear PDEs.

Contribution

The paper develops ultrafunctions using Non-Archimedean Mathematics to provide solutions to problems lacking classical or distributional solutions.

Findings

Ultrafunctions allow solutions to certain nonlinear PDEs without classical solutions.

Application of Non-Standard Analysis techniques in constructing ultrafunctions.

Potential for ultrafunctions to address broader classes of problems in analysis.

Abstract

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=u^{p-1}, u>0, p\geq(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.