Regularity of Lipschitz free boundaries for the thin one-phase problem

Abstract

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$ E(u,\Omega) = \int\_\Omega |\nabla u|^2 dX + \mathcal{H}^n({u>0} \cap {x\_{n+1} = 0}), \quad \Omega \subset \mathbb R^{n+1}, $$ among all functions $u\ge 0$ which are fixed on $\partial \Omega$.