Lower semicontinuity of nonlocal $L^\infty $ energies on $SBV_0(I)$

José Matias; Pedro Miguel Santos; Elvira Zappale

doi:10.5802/crmath.726

Article de recherche - Équations aux dérivées partielles

Lower semicontinuity of nonlocal

L^{\infty}

energies on

S B V_{0} (I)

[Semi-continuité inférieure des énergies

L^{\infty}

non locales sur

S B V_{0} (I)

]

José Matias ¹ ; Pedro Miguel Santos ¹ ; Elvira Zappale ^2,³

¹ Departamento de Matemática, Instituto Superior Técnico, University of Lisbon, Portugal
² Department of Basic and Applied Sciences for Engineering, Sapienza-University of Rome, via A. Scarpa 16, 00161 Roma, Italy
³ CIMA, Universidade de Évora, Portugal

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 407-414.

Abstract (VO)
Résumé

We characterize the lower-semicontinuity of nonlocal one-dimensional energies of the type

{ess \sup}_{(s, t) \in I \times I} h ([u] (s), [u] (t)),

where $I$ is an open and bounded interval in the real line, $u \in {SBV}_{0} (I)$ and $[u] (r) := u (r^{+}) - u (r^{-})$ , with $r \in I$ .

Nous caractérisons la semi-continuité inférieure des énergies non locales unidimensionnelles

{ess \sup}_{(s, t) \in I \times I} h ([u] (s), [u] (t)),

où I est un intervalle ouvert et borné dans l’axe réel, $u \in {SBV}_{0} (I)$ et $[u] (r) := u (r^{+}) - u (r^{-})$ avec $r \in I$ .

Reçu le : 2025-01-04
Accepté le : 2025-01-29
Accepté après révision le : 2025-02-04
Publié le : 2025-05-20

DOI : 10.5802/crmath.726

Classification : 49J45, 26B25, 26A45, 26A51
Keywords: Cartesian submaximality, nonlocal

L^{\infty}

energies, interfacial energy, lower semicontinuity, nonlocal gradients, supremal functionals
Mots-clés : Sous-maximalité cartésienne, énergies

L^{\infty}

non locales, énergie interfaciale, semi-continuité inférieure, gradients non locaux, fonctionnelles suprémales

Affiliations des auteurs :

José Matias ¹ ; Pedro Miguel Santos ¹ ; Elvira Zappale ^{2, 3}

¹ Departamento de Matemática, Instituto Superior Técnico, University of Lisbon, Portugal
² Department of Basic and Applied Sciences for Engineering, Sapienza-University of Rome, via A. Scarpa 16, 00161 Roma, Italy
³ CIMA, Universidade de Évora, Portugal

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2025__363_G4_407_0,
     author = {Jos\'e Matias and Pedro Miguel Santos and Elvira Zappale},
     title = {Lower semicontinuity of nonlocal $L^\infty $ energies on $SBV_0(I)$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {407--414},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.726},
     language = {en},
}

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José Matias; Pedro Miguel Santos; Elvira Zappale. Lower semicontinuity of nonlocal $L^\infty $ energies on $SBV_0(I)$. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 407-414. doi : 10.5802/crmath.726. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.726/

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