Uniform regularity and weight estimates for the Poisson problems when rounding off the conical points

Benoît Daniel; Simon Labrunie; Victor Nistor

doi:10.5802/crmath.770

Article de recherche - Équations aux dérivées partielles

Uniform regularity and weight estimates for the Poisson problems when rounding off the conical points
[Régularité uniforme et estimations de poids pour les problèmes de Poisson lors de l’arrondissement des points coniques]

Benoît Daniel ¹ ; Simon Labrunie ¹ ; Victor Nistor ¹

¹ Université de Lorraine, CNRS, IECL, 54000 Nancy, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1013-1023

Abstract (VO)
Résumé

We establish uniform solvability estimates for the Poisson problems associated to a suitably bounded family $\lbrace \Omega _{n}\rbrace _{n \in \mathfrak{I}}$ of domains in $\mathbb{R}^{d}$. The main example is that of a suitable sequence of smooth domains that “converges” to a domain with conical points by rounding off the conical points. We give full details for the case of a straight polygonal domain approximated by a sequence of smooth domains rounding off its corners. The method of proof relies on a conformal modification of the metric, with respect to which the union of our domains becomes a manifold with boundary and relative bounded geometry.

Nous établissons des estimées de résolubilité uniforme pour les problèmes de Poisson associés à une famille $\lbrace \Omega _{n}\rbrace _{n \in \mathfrak{I}}$ de domaines de $\mathbb{R}^{d}$ bornée de manière appropriée. L’exemple principal est celui d’une suite appropriée de domaines lisses qui « converge » vers un domaine à points coniques en arrondissant les points coniques. Nous donnons les détails complets dans le cas d’un domaine polygonal rectiligne approché par une suite de domaines lisses arrondissant ses angles. La méthode de la démonstration repose sur une modification conforme de la métrique, pour laquelle l’union de nos domaines devient une variété à bord à géométrie relativement bornée.

Reçu le : 2024-09-12
Révisé le : 2025-04-28
Accepté le : 2025-06-16
Publié le : 2025-09-05

DOI : 10.5802/crmath.770

Classification : 35R01, 35J15, 46E35, 58J32
Keywords: Strongly elliptic operators, Poincaré inequality, polygonal domain, Babuška–Kondratiev spaces, Sobolev spaces, manifolds with bounded geometry
Mots-clés : Opérateurs fortement elliptiques, inégalité de Poincaré, domaine polygonal, espaces de Babuška–Kondratiev, espaces de Sobolev, variétés à géométrie bornée

Affiliations des auteurs :

Benoît Daniel ¹ ; Simon Labrunie ¹ ; Victor Nistor ¹

¹ Université de Lorraine, CNRS, IECL, 54000 Nancy, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Beno{\^\i}t Daniel and Simon Labrunie and Victor Nistor},
     title = {Uniform regularity and weight estimates for the {Poisson} problems when rounding off the conical points},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1013--1023},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.770},
     language = {en},
}

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Benoît Daniel; Simon Labrunie; Victor Nistor. Uniform regularity and weight estimates for the Poisson problems when rounding off the conical points. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1013-1023. doi: 10.5802/crmath.770

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