A simplified counterexample to the integral representation of the relaxation of double integrals
We show that the lower-semicontinuous envelope of a non-convex double integral may not admit a representation as a double integral. By taking an integrand with value $+\infty $ except at three points (say $-1$, $0$ and $1$) we give a simple proof and an explicit formula for the relaxation that hopef...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.558/ |
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| author | Braides, Andrea |
| author_facet | Braides, Andrea |
| author_sort | Braides, Andrea |
| collection | DOAJ |
| description | We show that the lower-semicontinuous envelope of a non-convex double integral may not admit a representation as a double integral. By taking an integrand with value $+\infty $ except at three points (say $-1$, $0$ and $1$) we give a simple proof and an explicit formula for the relaxation that hopefully may shed some light on this type of problems. This is a simplified version of examples by Mora-Corral and Tellini, and Kreisbeck and Zappale, who characterize the lower-semicontinuous envelope via Young measures. |
| format | Article |
| id | doaj-art-3a6dc66e03af475d899d50397d7149a5 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-3a6dc66e03af475d899d50397d7149a52025-02-07T11:21:12ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G548749110.5802/crmath.55810.5802/crmath.558A simplified counterexample to the integral representation of the relaxation of double integralsBraides, Andrea0SISSA, via Bonomea 265, Trieste, ItalyWe show that the lower-semicontinuous envelope of a non-convex double integral may not admit a representation as a double integral. By taking an integrand with value $+\infty $ except at three points (say $-1$, $0$ and $1$) we give a simple proof and an explicit formula for the relaxation that hopefully may shed some light on this type of problems. This is a simplified version of examples by Mora-Corral and Tellini, and Kreisbeck and Zappale, who characterize the lower-semicontinuous envelope via Young measures.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.558/ |
| spellingShingle | Braides, Andrea A simplified counterexample to the integral representation of the relaxation of double integrals Comptes Rendus. Mathématique |
| title | A simplified counterexample to the integral representation of the relaxation of double integrals |
| title_full | A simplified counterexample to the integral representation of the relaxation of double integrals |
| title_fullStr | A simplified counterexample to the integral representation of the relaxation of double integrals |
| title_full_unstemmed | A simplified counterexample to the integral representation of the relaxation of double integrals |
| title_short | A simplified counterexample to the integral representation of the relaxation of double integrals |
| title_sort | simplified counterexample to the integral representation of the relaxation of double integrals |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.558/ |
| work_keys_str_mv | AT braidesandrea asimplifiedcounterexampletotheintegralrepresentationoftherelaxationofdoubleintegrals AT braidesandrea simplifiedcounterexampletotheintegralrepresentationoftherelaxationofdoubleintegrals |