The Dirichlet problem for second order parabolic operators in divergence form

Pascal Auscher; Moritz Egert; Kaj Nyström

doi:10.5802/jep.74

The Dirichlet problem for second order parabolic operators in divergence form
[Le problème de Dirichlet pour les opérateurs paraboliques sous forme divergence]

Pascal Auscher ¹ ; Moritz Egert ² ; Kaj Nyström ³

¹ Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay, France and Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR 7352 du CNRS, Université de Picardie-Jules Verne 80039 Amiens, France
² Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay, France
³ Department of Mathematics, Uppsala University S-751 06 Uppsala, Sweden

Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 407-441.

Résumé
Abstract

Nous étudions des opérateurs paraboliques $ℋ = \partial_{t} - {div}_{λ, x} A (x, t) \nabla_{λ, x}$ dans le demi-espace supérieur parabolique $ℝ_{+}^{n + 2} = {(λ, x, t) : λ > 0}$ . Nous supposons que les coefficients sont réels, bornés, mesurables, uniformément elliptiques, mais pas nécessairement symétriques. Nous montrons que la mesure parabolique associée est absolument continue par rapport à la mesure de surface sur $ℝ^{n + 1}$ au sens défini par $A_{\infty} (d x d t)$ . Notre argument donne aussi une preuve simplifiée du résultat correspondant pour la mesure elliptique.

We study parabolic operators $ℋ = \partial_{t} - {div}_{λ, x} A (x, t) \nabla_{λ, x}$ in the parabolic upper half space $ℝ_{+}^{n + 2} = {(λ, x, t) : λ > 0}$ . We assume that the coefficients are real, bounded, measurable, uniformly elliptic, but not necessarily symmetric. We prove that the associated parabolic measure is absolutely continuous with respect to the surface measure on $ℝ^{n + 1}$ in the sense defined by $A_{\infty} (d x d t)$ . Our argument also gives a simplified proof of the corresponding result for elliptic measure.

Reçu le : 2017-06-29
Accepté le : 2018-04-04
Publié le : 2018-04-26

MR Zbl

DOI : 10.5802/jep.74

Classification : 35K10, 35K20, 26A33, 42B25
Keywords: Second order parabolic operator, non-symmetric coefficients, Dirichlet problem, parabolic measure, $A_\infty $-condition, Carleson measure estimate.
Mot clés : Opérateur parabolique du second ordre, coefficients non symétriques, problème de Dirichlet, mesure parabolique, condition $A_\infty $, estimée de la mesure de Carlson.

Affiliations des auteurs :

Pascal Auscher ¹ ; Moritz Egert ² ; Kaj Nyström ³

¹ Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay, France and Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, UMR 7352 du CNRS, Université de Picardie-Jules Verne 80039 Amiens, France
² Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay, France
³ Department of Mathematics, Uppsala University S-751 06 Uppsala, Sweden

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Pascal Auscher and Moritz Egert and Kaj Nystr\"om},
     title = {The {Dirichlet} problem for second order parabolic operators in divergence form},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {407--441},
     publisher = {\'Ecole polytechnique},
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Pascal Auscher; Moritz Egert; Kaj Nyström. The Dirichlet problem for second order parabolic operators in divergence form. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 407-441. doi : 10.5802/jep.74. https://jep.centre-mersenne.org/articles/10.5802/jep.74/

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