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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 typefree boundary problem at the non-flat points and the related regularity of freeboundary. There, we also defined the functional$$\phi_p(r,u,x_0)=\frac1{r^4}\int_{B_r(x_0)}\frac{|\nablau^+(x)|^p}{|x-x_0|^{N-2}}dx\int_{B_r(x_0)}\frac{|\nablau^-(x)|^p}{|x-x_0|^{N-2}}dx$$ where $x_0$ is a free boundary point, i.e.$x_0\in\partial\{u>0\}$ and $u$ is a minimizer of the functional$$J(u):=\int_{\Omega}|\nabla u|^p +\lambda_+^p\,\chi_{\{u>0\}}+\lambda_-^p\,\chi_{\{u\le 0\}}, $$ for some bounded smooth domain$\Omega\subset {\mathbb R}^N$ and positive constants $\lambda_\pm$ with$\Lambda:=\lambda_+^p-\lambda^p_->0$.  Here we show the discrete monotonicity of $\phi_p(r,u,x_0)$ in two spatialdimensions at non-flat points, when $p$ is sufficiently close to 2, and thenestablish the linear growth. A new feature of our approach is the anisotropicscaling argument discussed in Section 4.

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