The Horizontal One-Phase Bernoulli Problem

The regularity of minimizers for the classical one-phase Bernoulli functional has been extensively studied following the pioneering work of Alt and Caffarelli. More recently, the regularity of almost minimizers has also been investigated. This relaxed notion of minimality arises naturally in various contexts, such as variational problems with constraints, and its flexibility allows for addressing a broader range of questions. We present some results concerning the regularity of almost minimizers for a one-phase Bernoulli-type functional in Carnot groups of step two. Our approach is inspired by the methods introduced by De Silva and Savin in the Euclidean setting. We outline the main challenges and techniques employed to establish local Lipschitz regularity for almost minimizers, emphasizing the role of intrinsic gradient estimates. Additionally, we discuss certain generalizations to nonlinear counterparts. Some of the results presented stem from joint works with F. Ferrari (University of Bologna) and N. Forcillo (Michigan State University).