The Dirichlet problem for second order parabolic operators in divergence form

Auscher, Pascal; Egert, Moritz; Nyström, Kaj

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

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

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

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.

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