A new elliptic measure on lower dimensional sets

TL;DR

This paper introduces a new elliptic measure for lower-dimensional sets, extending classical PDE and geometric measure theory results to sets with boundaries of lower codimension.

Contribution

It defines a novel class of degenerate elliptic operators and establishes initial results on the absolute continuity of the associated elliptic measure.

Findings

Elliptic measure is absolutely continuous with respect to surface measure.

The approach generalizes classical results to lower codimension boundaries.

Provides foundational steps for a PDE-based theory of lower-dimensional sets.

Abstract

The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of $n-1$ dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.