Describing limits of integrable functions as grid functions of nonstandard analysis

Emanuele Bottazzi ()
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Emanuele Bottazzi: University of Pavia

Partial Differential Equations and Applications, 2021, vol. 2, issue 4, 1-25

Abstract: Abstract In functional analysis, there are different notions of limit for a bounded sequence of $$L^1$$ L 1 functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of $$L^1$$ 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 nonstandard analysis, we show that for every bounded sequence $$\{z_n\}_{n \in \mathbb {N}}$$ { z n } n ∈ N of $$L^1$$ 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.

Keywords: Generalized functions; Nonstandard analysis; Nonlinear ill-posed problems; 46F30; 46S20; 47J06; 35K55 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s42985-021-00093-9

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