Extendability of functions with partially vanishing trace

Bechtel, Sebastian; Brown, Russell M.; Haller, Robert; Tolksdorf, Patrick

doi:10.5802/aif.3707

Extendability of functions with partially vanishing trace
[Prolongement des fonctions qui disparaissent sur une partie du bord]

Bechtel, Sebastian ¹ ; Brown, Russell M.² ; Haller, Robert ³ ; Tolksdorf, Patrick ⁴

¹ Delft Institute of Applied Mathematics Delft University of Technology P.O. Box 5031 2600 GA Delft (The Netherlands)
² Department of Mathematics POT 715 University of Kentucky Lexington, KY 40506-0027 (USA)
³ Fachbereich Mathematik Technische Universität Darmstadt Schlossgartenstr. 7 64289 Darmstadt (Germany)
⁴ Faculty of Mathematics Karlsruhe Institute of Technology Englerstr. 2 76131 Karlsruhe (Germany)

Annales de l'Institut Fourier, Online first, 49 p.

Résumé
Abstract

Soit $Ω \subseteq ℝ^{d}$ ouvert et $D \subseteq \partial Ω$ une partie fermée du bord. Sous des hypothèses faibles sur $Ω$ , nous construisons un opérateur de prolongement borné pour l’espace Sobolev $W_{D}^{k, p} (Ω)$ , $1 \leq p < \infty$ , ce qui est le sous-espace de $W^{k, p} (Ω)$ composé des fonctions qui disparaissent sur $D$ dans un sens approprié. Au contraire des travaux précédents, notre construction est globale et ne se sert pas d’arguments de localisation, ce qui nous permet de travailler avec une condition de régularité exacte pour l’interface entre $D$ et $\partial Ω ∖ D$ . Aussi, nous fournissons des estimations homogènes et locales pour cet opérateur de prolongement. En plus, nous traitons le cas de fonctions Lipschitz disparaissant sur $D$ .

Let $Ω \subseteq ℝ^{d}$ be open and $D \subseteq \partial Ω$ be a closed part of its boundary. Under very mild assumptions on $Ω$ , we construct a bounded Sobolev extension operator for the Sobolev space $W_{D}^{k, p} (Ω)$ , $1 \leq p < \infty$ , which consists of all functions in $W^{k, p} (Ω)$ that vanish in a suitable sense on $D$ . In contrast to earlier work, this construction is global and does not use a localization argument, which allows to work with a boundary regularity that is sharp at the interface dividing $D$ and $\partial Ω ∖ D$ . Moreover, we provide homogeneous and local estimates for the extension operator. Also, we treat the case of Lipschitz function spaces with a vanishing trace condition on $D$ .

Reçu le : 2021-04-27
Révisé le : 2022-12-13
Accepté le : 2023-10-02
Première publication : 2025-02-10

DOI : 10.5802/aif.3707

Classification : 46E35, 35J58, 46A22
Keywords: Sobolev extension operators, mixed boundary value problems, $(\varepsilon , \delta )$-domains
Mots-clés : Opérateur de prolongement pour des espaces Sobolev, conditions aux limites mixtes, $(\varepsilon , \delta )$-domaines

Affiliations des auteurs :

Bechtel, Sebastian ¹ ; Brown, Russell M. ² ; Haller, Robert ³ ; Tolksdorf, Patrick ⁴

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     title = {Extendability of functions with partially vanishing trace},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     doi = {10.5802/aif.3707},
     language = {en},
     note = {Online first},
}

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Bechtel, Sebastian; Brown, Russell M.; Haller, Robert; Tolksdorf, Patrick. Extendability of functions with partially vanishing trace. Annales de l'Institut Fourier, Online first, 49 p.

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