Regularity of almost minimizers of horizontal one-phase Bernoulli problems

The thesis mainly concerns the study of the regularity of almost minimizers of some one-phase Bernoulli-type functionals in Carnot Groups of step two. We consider some intrinsic functionals that represents the natural counterpart in Carnot groups to the classical one-phase Bernuolli functional studied in the Euclidean setting by Alt and Caffarelli. Generalizing the approach of De Silva and Savin, the main contribution of this thesis stems from the local intrinsic Lipschitz regularity of almost minimizers. Moreover, we extend the intrinsic Lipschitz regularity of almost minimizers to the nonlinear context of the p-Bernoulli-type functionals, driven by the sub-p-Laplace operator.