Abstract
AbstractIn 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.
Funder
Università degli Studi di Pavia
Publisher
Springer Science and Business Media LLC
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