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

José Matias; Pedro Miguel Santos; Elvira Zappale

doi:10.5802/crmath.726

Research article - Partial differential equations

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

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 (Original language)
Résumé

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

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

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

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

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

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

Received: 2025-01-04
Accepted: 2025-01-29
Revised after acceptance: 2025-02-04
Published online: 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

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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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