Minimizers for the Thin One‐Phase Free Boundary Problem

Abstract We consider the “thin one‐phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of function in plus area positivity set that . establish full regularity for dimensions , prove almost everywhere arbitrary dimension, and provide content structure estimates on singular when it exists. All these results hold range relevant weight. While our are typical calculus variations, approach does not follow standard one first introduced by Alt Caffarelli 1981. Instead, nonlocal nature distributional measure minimizer necessitates arguments less reliant underlying PDE. © 2021 Wiley Periodicals LLC.