Harmonic measure on sets of codimension larger than one

TL;DR

This paper introduces a new harmonic measure for sets with codimension greater than one, linked to a degenerate elliptic PDE, establishing absolute continuity with respect to Hausdorff measure for certain Lipschitz sets.

Contribution

It defines a novel harmonic measure for higher codimension sets and develops an elliptic theory demonstrating its absolute continuity properties.

Findings

Harmonic measure is well-defined for codimension > 1 sets.

The measure is absolutely continuous with respect to Hausdorff measure on Lipschitz graphs.

Provides foundational results for further analysis of higher codimension sets.

Abstract

We introduce a new notion of a harmonic measure for a $d$-dimensional set in $\R^n$ with $d<n-1$, that is, when the codimension is strictly bigger than 1. Our measure is associated to a degenerate elliptic PDE, it gives rise to a comprehensive elliptic theory, and, most notably, it is absolutely continuous with respect to the $d$-dimensional Hausdorff measure on reasonably nice sets. This note provides general strokes of the proof of the latter statement for Lipschitz graphs with small Lipschitz constant.