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Hessian regularity in Hölder spaces for a semi-linear bi-Laplacian equation
[Régularité du hessien dans les espaces de Hölder pour une équation bi-laplacienne semi-linéaire]
Claudemir Alcantara1 ; Edgard Pimentel2 ; José Miguel Urbano2, 3
1 Department of Mathematics, Pontifical Catholic University of Rio de Janeiro (PUC-Rio), 22451-900 Rio de Janeiro, Brazil
2 CMUC, Department of Mathematics, University of Coimbra, 3000-143 Coimbra, Portugal
3 Applied Mathematics and Computational Sciences Program (AMCS), Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955 -6900, Kingdom of Saudi Arabia
Comptes Rendus. Mathématique, Volume 363 (2025), pp. 533-539.

We examine a semi-linear variant of the bi-Laplacian equation in the superlinear, subquadratic setting and obtain C2,σ-regularity estimates, depending on the growth regime of the nonlinearity. Our strategy is to render this fourth-order problem as a system of two Poisson equations and explore the interplay between the integrability and smoothness available for each equation taken isolated.

Nous examinons une variante semi-linéaire de l’équation bi-laplacienne dans un cadre superlinéaire et subquadratique, et obtenons des estimations de régularité C2,σ dépendant du régime de croissance de la non-linéarité. Notre stratégie consiste à transformer le problème du quatrième ordre en un système de deux équations de Poisson, puis à explorer l’interaction entre l’intégrabilité et la régularité disponibles pour chaque équation prise isolément.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.737
Classification : 35B65, 35J91, 35G20
Keywords: Bi-Laplacian operator, semi-linear equations, Hessian regularity, Hölder spaces
Mots-clés : Opérateur bi-laplacien, équations semi-linéaires, régularité hessienne, espaces de Hölder

Claudemir Alcantara 1 ; Edgard Pimentel 2 ; José Miguel Urbano 2, 3

1 Department of Mathematics, Pontifical Catholic University of Rio de Janeiro (PUC-Rio), 22451-900 Rio de Janeiro, Brazil
2 CMUC, Department of Mathematics, University of Coimbra, 3000-143 Coimbra, Portugal
3 Applied Mathematics and Computational Sciences Program (AMCS), Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955 -6900, Kingdom of Saudi Arabia
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Claudemir Alcantara and Edgard Pimentel and Jos\'e Miguel Urbano},
     title = {Hessian regularity in {H\"older} spaces for a semi-linear {bi-Laplacian} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {533--539},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.737},
     language = {en},
}
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Claudemir Alcantara; Edgard Pimentel; José Miguel Urbano. Hessian regularity in Hölder spaces for a semi-linear bi-Laplacian equation. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 533-539. doi : 10.5802/crmath.737. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.737/

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