Closure result for $\Gamma$-limits of functionals with linear growth

Jesenko, Martin (author)

We consider integral functionals $F_\epsilon^{(j)}$, doubly indexed by $\epsilon$ > 0 and $j \in \mathbb{N} \cup \{\infty\}$, satisfying a standard linear growth condition. We investigate the question of $\Gamma$-closure, i.e., when the $\Gamma$-convergence of all families $\{F_\epsilon^{(j)}\}_ε$ with finite $j$ implies $\Gamma$-convergence of $\{F_\epsilon^{(\infty)}\}_ε$. This has already been explored for $p$-growth with $p$ > 1. We show by an explicit counterexample that due to the differences between the spaces $W^{1,1}$ and $W^{1,p}$ with $p$ > 1, an analog cannot hold. Moreover, we find a sufficient condition for a positive answer.

• Language: English
• Keywords: $\Gamma$-convergence, $\Gamma$-closure results, functionals with standard linear growth
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FGG - Faculty of Civil and Geodetic Engineering
• Publication status: Published
• Publication version: Version of Record
• Year: 2023
• Number of pages: Str. 2333–2343
• Numbering: Vol. 202, iss. 5
• PID: 20.500.12556/RUL-151929
• UDC: 517
• ISSN on article: 0373-3114
• DOI: 10.1007/s10231-023-01322-1
• COBISS.SI-ID: 153548803
• Publication date in RUL: 25.10.2023

Record is a part of a journal

• Title: Annali di matematica pura ed applicata
• Shortened title: Ann. mat. pura appl.
• Publisher: Springer Nature, Fondazione Annali di Matematica Pura ed Applicata
• ISSN: 0373-3114
• COBISS.SI-ID: 24962816

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: gama konvergenca, gama zaprtje, funkcionali s standardno linearno rastjo