# Minimizers for the thin one-phase free boundary problem

## TL;DR

This paper studies the regularity and structure of the free boundary in the thin one-phase free boundary problem, establishing full regularity in low dimensions and near-everywhere regularity in higher dimensions, using a novel approach that handles nonlocal measures.

## Contribution

It provides new regularity results for the free boundary in the thin one-phase problem, especially in higher dimensions, with a nonstandard approach for nonlocal measures.

## Key findings

- Full regularity of the free boundary for n ≤ 2
- Almost everywhere regularity in arbitrary dimensions
- Content and structure estimates on the singular set

## Abstract

We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb R^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb R^n$. We establish full regularity of the free boundary for dimensions $n \leq 2$, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced in \cite{AltCaffarelli}. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE.