| Issue |
ESAIM: COCV
Volume 27, 2021
|
|
|---|---|---|
| Article Number | 52 | |
| Number of page(s) | 24 | |
| DOI | https://doi.org/10.1051/cocv/2021048 | |
| Published online | 04 June 2021 | |
on a certain compactification of an arbitrary subset of ℝm and its applications to DiPerna-Majda measures theory
Military University of Technology,
ul. Gen. Sylwestra Kaliskiego 2,
00-908
Warszawa, Poland.
* Corresponding author: piotr.a.k.kozarzewski@gmail.com
Received:
22
June
2020
Accepted:
25
April
2021
We present a constructive proof of the fact, that for any subset 𝒜 ⊆ ℝm 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 f̅ : 𝒜′→ℝ. 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(uν)} 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.
Mathematics Subject Classification: 49J45 / 49R99 / 28A33
Key words: DiPerna-Majda measures / compactification / weak-⋆ convergence
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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