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

### Daniela De Silva

Columbia University, New York, USA### Ovidiu Savin

Columbia University, New York, United States

## 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 $\partial \Omega$.

## Cite this article

Daniela De Silva, Ovidiu Savin, Regularity of Lipschitz free boundaries for the thin one-phase problem. J. Eur. Math. Soc. 17 (2015), no. 6, pp. 1293–1326

DOI 10.4171/JEMS/531