Uniform Lipschitz regularity for classes of minimizers in two phase free boundary problems in Orlicz spaces with small density on the negative phase

In this paper we investigate Lipschitz regularity of minimizers for classes of functionals including ones of the type EG(u,Ω)=∫Ω[G(|∇u|)+f2χ{u>0}+f1χ{u⩽0}]dx. We prove that there exists a universal “tolerance” (depending only on the degenerate ellipticity and other intrinsic parameters) for the density of the negative phase along the free boundary under which uniform Lipschitz regularity holds. We also prove density estimates from below for the negative phase on points inside the contact set between the negative and positive free boundaries in the case where Lipschitz regularity fails to be the optimal one.