The Dirichlet problem for second order parabolic operators in divergence form

Overview & Synopsis

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.