The Dirichlet problem for second order parabolic operators in divergence form

TL;DR

This paper proves the absolute continuity of parabolic measure for second order divergence form operators with measurable coefficients in the upper half space, extending results to non-symmetric cases and providing a simplified elliptic measure proof.

Contribution

It establishes absolute continuity of parabolic measure for non-symmetric divergence form operators, generalizing previous symmetric cases and simplifying elliptic measure proofs.

Findings

Parabolic measure is absolutely continuous with respect to surface measure.

Results hold for non-symmetric, measurable, uniformly elliptic coefficients.

Provides a simplified proof for elliptic measure case.

Abstract

We study parabolic operators H = $\partial$t -- 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$\infty$(dx dt). Our argument also gives a simplified proof of the corresponding result for elliptic measure.