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

Abstract

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 $${\varphi }_p(r,u,x_0)=\frac{1}{r^4}{\int }_{B_r(x_0)}\frac{|\nabla u^{+}(x){|}^p}{|x-x_0{|}^{N-2}}dx{\int }_{B_r(x_0)}\frac{|\nabla u^{-}(x){|}^p}{|x-x_0{|}^{N-2}}dx$$ 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 ${\varphi }_p(r,u,x_0)$ 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.