Square function/non-tangential maximal function estimates and the dirichlet problem for non-symmetric elliptic operators

TL;DR

This paper proves square function and non-tangential maximal function estimates for solutions to non-symmetric elliptic operators in half-space, establishing absolute continuity of harmonic measure and solvability of the Dirichlet problem in higher dimensions.

Contribution

It extends known results from one-dimensional cases to higher dimensions for non-symmetric elliptic operators, linking estimates to boundary measure properties.

Findings

Harmonic measure is in A_infinity class with respect to surface measure.

Dirichlet problem is solvable with L^p data for some p<.

Square function and non-tangential maximal estimates are established for solutions.

Abstract

We consider divergence form elliptic operators L = - div A(x)\nabla, defined in the half space R^{n+1}_+, n \geq 2, where the coefficient matrix A(x) is bounded, measurable, uniformly elliptic, t-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation Lu = 0, and we then combine these estimates with the method of "\epsilon-approximability" to show that L-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class A_\infty with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in L^p, for some p < \infty). Previously, these results had been known only in the case n = 1.