The Dirichlet problem for second order parabolic operators in divergence form

Pascal Auscher,Kaj Nyström,Moritz Egert

doi:10.5802/jep.74

Abstract

We study parabolic operators H = ∂t − div λ,x A(x, t)∇ 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∞(dx dt). Our argument also gives a simplified proof of the corresponding result for elliptic measure.

Highlights

Introduction and statement of main resultsA classical result due to Dahlberg [8] states in the context of Lipschitz domains that harmonic measure is absolutely continuous with respect to surface measure, and that the Poisson kernel satisfies a scale-invariant reverse Hölder inequality in L2
The Dirichlet problem with L2-data can be solved with L2-control of a non-tangential maximal function
In [22, 21] the mutual absolute continuity of parabolic measure and surface measure and the A∞-property were established and in [16] the authors obtained a version of Dahlberg’s result for parabolic measure associated to the heat equation in time-dependent Lipschitz-type domains. In this context the properties of parabolic measures were further analyzed in the influential work [15], parts of which have been simplified in [28]

Summary

Introduction and statement of main results

A classical result due to Dahlberg [8] states in the context of Lipschitz domains that harmonic measure is absolutely continuous with respect to surface measure, and that the Poisson kernel (its Radon-Nikodym derivative) satisfies a scale-invariant reverse Hölder inequality in L2. — The key insight in [19] is that the A∞-property of elliptic measure follows once a certain Carleson measure condition is verified This idea has been implemented in the parabolic context: On pp.1172–1175 in [9] it is shown that in order to conclude ω ∈ A∞(dx dt) it suffices to prove the following result, which we state here as our second main theorem. — Theorem 1.6 is a priori equivalent to the statement that (1.5) holds for all parabolic cubes whenever u is the unique solution to the continuous Dirichlet problem for H u = 0 with continuous compactly supported boundary data f satisfying |f | 1, see [9, Rem. 5] Note that in this case |u| 1 by the maximum principle.

Technical tools

Functional calculus and square function estimates

Non-tangential maximal function estimates

Parabolic sawtooth domains associated with F

Proof of the Key Lemma