Harmonic measure on sets of codimension larger than one

Guy David; Joseph Feneuil; Svitlana Mayboroda

doi:10.1016/j.crma.2017.02.013

Mathematical analysis/Harmonic analysis

Harmonic measure on sets of codimension larger than one
[Mesure harmonique sur un ensemble de dimension supérieure à 1]

Présenté par : Gilles Pisier

Guy David ¹ ; Joseph Feneuil ² ; Svitlana Mayboroda ²

¹ Université Paris-Sud, Laboratoire de mathématiques, UMR 8628, 91405 Orsay, France
² School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 406-410.

Résumé
Abstract

On introduit une nouvelle notion de mesure harmonique sur un ensemble $Γ \subset R^{n}$ Ahlfors-régulier de dimension $d < n - 1$ . Notre mesure est associée à un opérateur différentiel linéaire elliptique dégénéré L, et a les mêmes propriétés générales qu'en codimension 1 (mesure doublante, principe de comparaison pour les fonctions L-harmoniques positives). De plus, elle est absolument continue par rapport à la mesure de Hausdorff de dimension d dans des cas simples. Cette note décrit la démonstration des propriétés générales et de l'absolue continuité quantifiée quand Γ est un petit graphe lipschitzien et L est bien choisi.

We introduce a new notion of a harmonic measure for a d-dimensional set in $R^{n}$ with $d < n - 1$ , that is, when the codimension is strictly bigger than 1. Our measure is associated with a degenerate elliptic PDE, it gives rise to a comprehensive elliptic theory, and, most notably, it is absolutely continuous with respect to the d-dimensional Hausdorff measure on reasonably nice sets. This note provides general strokes of the proof of the latter statement for Lipschitz graphs with small Lipschitz constant.

Reçu le : 2016-08-06
Accepté le : 2017-02-28
Publié le : 2017-03-14

DOI : 10.1016/j.crma.2017.02.013

Affiliations des auteurs :

Guy David ¹ ; Joseph Feneuil ² ; Svitlana Mayboroda ²

¹ Université Paris-Sud, Laboratoire de mathématiques, UMR 8628, 91405 Orsay, France
² School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

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     title = {Harmonic measure on sets of codimension larger than one},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {406--410},
     publisher = {Elsevier},
     volume = {355},
     number = {4},
     year = {2017},
     doi = {10.1016/j.crma.2017.02.013},
     language = {en},
}

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Guy David; Joseph Feneuil; Svitlana Mayboroda. Harmonic measure on sets of codimension larger than one. Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 406-410. doi : 10.1016/j.crma.2017.02.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.02.013/

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Cité par Sources :

^☆ Svitlana Mayboroda is supported in part by the Alfred P. Sloan Fellowship, the NSF INSPIRE Award DMS 1344235, NSF CAREER Award DMS 1220089. Guy David is supported in part by the ANR, programme blanc GEOMETRYA ANR-12-BS01-0014. Joseph Feneuil is partially supported by the ANR project “HAB” No. ANR-12-BS01-0013. This work was supported by a public grant as part of the “Investissement d'avenir” project, reference ANR-11-LABX-0056-LMH, LabEx LMH. Part of this work was completed during S. Mayboroda's visit to Université Paris-Sud, Laboratoire de mathématiques, Orsay, and École polytechnique, PMC. We thank the corresponding Departments and Fondation Jacques-Hadamard for support and hospitality.

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