Describing limits of integrable functions as grid functions of nonstandard analysis

TL;DR

This paper explores how nonstandard analysis can describe various limits of bounded sequences of $L^1$ functions using hyperfinite grid functions, with applications to nonlinear PDEs.

Contribution

It demonstrates that hyperfinite grid functions in nonstandard analysis can simultaneously represent weak-$ star$ and Young measure limits of bounded $L^1$ sequences, providing a new analytical tool.

Findings

Hyperfinite grid functions represent both weak-$ star$ and Young measure limits.

Application to nonlinear PDEs, including ill-posed equations.

Bridges standard and nonstandard analysis in functional limits.

Abstract

In functional analysis, there are different notions of limit for a bounded sequence of $L^1$ functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of $L^1$ functions can be described in terms of its weak-$\star$ limit or by introducing a measure-valued notion of limit in the sense of Young measures. Working in Robinson's framework of analysis with infinitesimals, we show that for every bounded sequence $\{z_n\}_{n \in \mathbb{N}}$ of $L^1$ functions there exists a function of a hyperfinite domain (i.e.\ a grid function) that represents both the weak-$\star$ and the Young measure limits of the sequence. This result has relevant applications to the study of nonlinear PDEs. We discuss the example of an ill-posed forward-backward parabolic equation.