The Dirichlet problem for second order parabolic operators in divergence form
Journal de l’École polytechnique - Mathématiques, Volume 5 (2018), pp. 407-441.

We study parabolic operators $ℋ={\partial }_{t}-{\mathrm{div}}_{\lambda ,x}A\left(x,t\right){\nabla }_{\lambda ,x}$ in the parabolic upper half space ${ℝ}_{+}^{n+2}=\left\{\left(\lambda ,x,t\right):\phantom{\rule{4pt}{0ex}}\lambda >0\right\}$. 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 ${A}_{\infty }\left(\mathrm{d}x\phantom{\rule{0.166667em}{0ex}}\mathrm{d}t\right)$. Our argument also gives a simplified proof of the corresponding result for elliptic measure.

Nous étudions des opérateurs paraboliques $ℋ={\partial }_{t}-{\mathrm{div}}_{\lambda ,x}A\left(x,t\right){\nabla }_{\lambda ,x}$ dans le demi-espace supérieur parabolique ${ℝ}_{+}^{n+2}=\left\{\left(\lambda ,x,t\right):\phantom{\rule{4pt}{0ex}}\lambda >0\right\}$. 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 ${A}_{\infty }\left(\mathrm{d}x\phantom{\rule{0.277778em}{0ex}}\mathrm{d}t\right)$. Notre argument donne aussi une preuve simplifiée du résultat correspondant pour la mesure elliptique.

Accepted:
Published online:
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.
Auscher, Pascal 1; Egert, Moritz 2; Nyström, Kaj 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
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Auscher, Pascal; Egert, Moritz; Nyström, Kaj. The Dirichlet problem for second order parabolic operators in divergence form. Journal de l’École polytechnique - Mathématiques, Volume 5 (2018), pp. 407-441. doi : 10.5802/jep.74. http://archive.numdam.org/articles/10.5802/jep.74/

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