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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| Main Author: | |
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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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| Summary: | 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. |
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| ISSN: | 1778-3569 |