A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two

We continue the analysis of the two-phase free boundary problems initiated by ourselves, studying where we studied the linear growth of minimizers in a Bernoulli-type free boundary problem at the non-flat points and the related regularity of free boundary. There, among other things, we also defined the functional (Formula presented.) where x ₀ is a free boundary point, i.e. (Formula presented.) and u is a minimizer of the functional (Formula presented.) for some bounded smooth domain (Formula presented.) and positive constants (Formula presented.) with (Formula presented.). Here we show (Formula presented.) in discrete monotone at non-flat points x ₀ , when N = 2 and p is sufficiently close to 2, and then establish the linear growth of u. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.