On the Lipschitz regularity of solutions of a minimum problem with free boundary

In this article under the assumption of "small" density for the negativity set, we prove local Lipschitz regularity for the two-phase minimization problem with free boundary for the functional {equation presented} where λ₁, λ₂ are positive constants so that A = λp/1 - λp/2 < 0, xd is the characteristic function of the set D, Ω ⊂ ℝⁿ is a (smooth) domain and the minimum is taken over a suitable subspace of W ¹ p (Ω).