Parabolic NTA domains in $\mathbb R^2$

TL;DR

This paper proves that boundary components of parabolic NTA domains in two dimensions are graph-like and uses this to classify blowup solutions for a specific caloric measure free boundary problem.

Contribution

It establishes that boundary components are graphs and applies this to classify blowup solutions in $\

Findings

Boundaries of parabolic NTA domains are graphs.

Classification of blowup solutions for the caloric measure problem.

Enhanced understanding of free boundary regularity in 2D.

Abstract

We show that each connected component of the boundary of a parabolic NTA domain in $\mathbb R^2$ is given by a graph. We then apply this observation to classify blowup solutions in $\mathbb R^2$ to a free boundary problem for caloric measure first considered by Hofmann, Lewis and Nystr\"om (Duke Math J. 2004).