We present a constructive proof of the fact, that for any subset 𝒜 ⊆ ℝ$$ and a countable family ℱ of bounded functions f : 𝒜 → ℝ there exists a compactification 𝒜′ ⊂ ℓ2 of 𝒜 such that every function f ∈ ℱ possesses a continuous extension to a function $$. However related to some classical theorems, our result is direct and hence applicable in Calculus of Variations. Our construction is then used to represent limits of weakly convergent sequences {f(u$$)} via methods related to DiPerna-Majda measures. In particular, as our main application, we generalise the Representation Theorem from the Calculus of Variations due to Kałamajska.
Keywords: DiPerna-Majda measures, compactification, weak-⋆ convergence
@article{COCV_2021__27_1_A54_0,
author = {Kozarzewski, Piotr Antoni},
title = {On a certain compactification of an arbitrary subset of $\mathbb{R}^m$ and its applications to {DiPerna-Majda} measures theory},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021048},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021048/}
}
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AU - Kozarzewski, Piotr Antoni
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JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2021
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PB - EDP-Sciences
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Kozarzewski, Piotr Antoni. On a certain compactification of an arbitrary subset of $\mathbb{R}^m$ and its applications to DiPerna-Majda measures theory. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 52. doi: 10.1051/cocv/2021048
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