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Boundary regularity for the polyharmonic Dirichlet problem
[Régularité au bord pour le problème de Dirichlet polyharmonique]
Antoine Lemenant1 ; Rémy Mougenot2
1Institut Universitaire de France (IUF), Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, BP 70239 54506 Vandœuvre-lès-Nancy Cedex, France
2Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 3, pp. 743-783

In this paper we prove that any solution of the $m$-polyharmonic Poisson equation in a Reifenberg-flat domain with homogeneous Dirichlet boundary condition, is $\mathscr{C}^{m-1,\alpha }$ regular up to the boundary. To achieve this result we extend the Nirenberg method of translations to operators of arbitrary order, and then use some Mosco-convergence tools developped in a previous paper [12].

Dans cet article, nous démontrons que toute solution de l’équation de Poisson $m$-polyharmonique dans un domaine Reifenberg-plat, munie de conditions aux limites de Dirichlet homogènes, est de régularité $\mathscr{C}^{m-1,\alpha }$ jusqu’au bord. Pour obtenir ce résultat, nous étendons la méthode des translations de Nirenberg à des opérateurs d’ordre arbitraire, puis nous utilisons des outils de Mosco-convergence développés dans un article précédent [12].

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1862
Classification : 35B65, 49Q99
Keywords: Polyharmonic problem, elliptic regularity, Reifenberg-flat domain
Mots-clés : Problème Polyharmonique, régularité elliptique, domaine Reifenberg-plat

Antoine Lemenant  1   ; Rémy Mougenot  2

1 Institut Universitaire de France (IUF), Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, BP 70239 54506 Vandœuvre-lès-Nancy Cedex, France
2 Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, BP 70239, 54506 Vandœuvre-lès-Nancy Cedex, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Antoine Lemenant; Rémy Mougenot. Boundary regularity for the polyharmonic Dirichlet problem. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 3, pp. 743-783. doi: 10.5802/afst.1862
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     volume = {Ser. 6, 35},
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