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 in \cite{DK}, 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, we also defined the functional $${\varphi }_p(r,u,x_0)=\frac{1}{r^4}{\int }_{B_r(x_0)}\frac{\mathrm{\mid }\mathrm{\nabla }{u}^{+}(x){\mathrm{\mid }}^p}{\mathrm{\mid }x-x_0{\mathrm{\mid }}^{N-2}}dx{\int }_{B_r(x_0)}\frac{\mathrm{\mid }\mathrm{\nabla }{u}^{-}(x){\mathrm{\mid }}^p}{\mathrm{\mid }x-x_0{\mathrm{\mid }}^{N-2}}dx$$ where $x_0$ is a free boundary point, i.e. $x_0\in \mathrm{\partial }\{u>0\}$ and $u$ is a minimizer of the functional $$J(u):={\int }_{\mathrm{\Omega }}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p+{\lambda }_{+}^p{\chi }_{\{u>0\}}+{\lambda }_{-}^p{\chi }_{\{u\le 0\}},$$ for some bounded smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ and positive constants ${\lambda }_{\pm }$ with $\mathrm{\Lambda }:={\lambda }_{+}^p-{\lambda }_{-}^p>0$. Here we show the discrete monotonicity of ${\varphi }_p(r,u,x_0)$ in two spatial dimensions at non-flat points, when $p$ is sufficiently close to 2, and then establish the linear growth. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.