A simplified counterexample to the integral representation of the relaxation of double integrals

Andrea Braides

doi:10.5802/crmath.558

Research article - Partial differential equations

A simplified counterexample to the integral representation of the relaxation of double integrals

Andrea Braides ¹

¹ SISSA, via Bonomea 265, Trieste, Italy

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 487-491.

Abstract
Résumé

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.

Nous montrons que l’enveloppe semi-continue inférieure d’une intégrale double non convexe peut ne pas admettre de représentation sous forme d’intégrale double. En prenant un intégrande avec une valeur infinie sauf en trois points (disons -1, 0 et 1), nous donnons une preuve simple et une formule explicite pour la relaxation qui, espérons-le, pourra éclairer ce type de problèmes. Ceci est une version simplifiée des exemples de Mora-Corral et Tellini, et de Kreisbeck et Zappale, qui caractérisent l’enveloppe semi-continue inférieure via les mesures de Young.

Received: 2023-05-11
Accepted: 2023-08-30
Published online: 2024-05-31

DOI: 10.5802/crmath.558

Author's affiliations:

Andrea Braides ¹

¹ SISSA, via Bonomea 265, Trieste, Italy

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {A simplified counterexample to the integral representation of the relaxation of double integrals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {487--491},
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     year = {2024},
     doi = {10.5802/crmath.558},
     language = {en},
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Andrea Braides. A simplified counterexample to the integral representation of the relaxation of double integrals. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 487-491. doi : 10.5802/crmath.558. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.558/

References
Cited by

[1] J. C. Bellido; C. Mora-Corral Lower semicontinuity and relaxation via Young measures for nonlocal variational problems and applications to peridynamics, SIAM J. Math. Anal., Volume 50 (2018), pp. 779-809 | DOI | MR | Zbl

[2] E. De Giorgi; G. Buttazzo; G. Dal Maso On the lower semicontinuity of certain integral functionals, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat., Volume 74 (1983), pp. 274-282 | MR | Zbl

[3] C. Kreisbeck; E. Zappale Loss of double-integral character during relaxation, SIAM J. Math. Anal., Volume 53 (2021), pp. 351-385 | DOI | MR | Zbl

[4] C. Mora-Corral; A. Tellini Relaxation of a scalar nonlocal variational problem with a double-well potential, Calc. Var. Partial Differ. Equ., Volume 59 (2020), 67, 57, 30 pages | DOI | MR | Zbl

[5] P. Pedregal Weak lower semicontinuity and relaxation for a class of non-local functionals, Rev. Mat. Complut., Volume 29 (2016), pp. 485-495 | DOI | MR | Zbl

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