Generalized solutions of variational problems and applications

TL;DR

This paper introduces ultrafunctions, a new class of generalized functions on hyperreal fields, to solve variational problems lacking classical solutions, and explores their properties and applications.

Contribution

It constructs ultrafunctions and analyzes their relationship with classical solutions, demonstrating their effectiveness in solving previously unsolvable variational problems.

Findings

Ultrafunctions can find solutions where classical methods fail.

Relationship established between ultrafunction solutions and classical minimizing sequences.

Examples show the potential of ultrafunctions in variational analysis.

Abstract

Ultrafunctions are a particular class of generalized functions defined on a hyperreal field $\mathbb{R}^{*}\supset\mathbb{R}$ that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach.