Describing limits of integrable functions as grid functions of\n nonstandard analysis

2021/01/20 by Emanuele Bottazzi, Bottazzi, Emanuele
Mathematics · #46F30 46S20 47J06 35K55 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2101.08108

openalex publication_date 2021/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In functional analysis, there are different notions of limit for a bounded\nsequence of L¹ functions. Besides the pointwise limit, that does not always\nexist, the behaviour of a bounded sequence of L¹ functions can be described\nin terms of its weak-\⋆ limit or by introducing a measure-valued notion of\nlimit in the sense of Young measures. Working in Robinson's framework of\nanalysis with infinitesimals, we show that for every bounded sequence\n z_{n n \∈ \ℕ} of L¹ functions there exists a function of a\nhyperfinite domain (i.e. a grid function) that represents both the\nweak-\⋆ and the Young measure limits of the sequence. This result has\nrelevant applications to the study of nonlinear PDEs. We discuss the example of\nan ill-posed forward-backward parabolic equation.\n

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