Minimizers for the Thin One-Phase Free Boundary Problem

We consider the thin one-phase free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in Double-struck capital R+n+1 plus the area of the positivity set of that function in Double-struck capital Rn. We establish full regularity of the free boundary for dimensions n <= 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 by Alt and Caffarelli in 1981. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments that are less reliant on the underlying PDE. (c) 2021 Wiley Periodicals LLC.

Automated summary

The study focuses on the one-phase free boundary problem related to minimizing weighted Dirichlet energy and the area of the positivity set of a function. The research demonstrates full regularity of the free boundary in certain dimensions, almost everywhere regularity in arbitrary dimensions, and provides estimates on the singular set of the free boundary. The results are applicable across a range of relevant weights and present a novel approach distinct from standard methods.