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 $$ϕ_p(r,u,x_0)=\frac1{r^4}\int_{B_r(x_0)}\frac{|\nabla u^+(x)|^p}{|x-x_0|^{N-2}}dx\int_{B_r(x_0)}\frac{|\nabla u^-(x)|^p}{|x-x_0|^{N-2}}dx$$ where $x_0$ is a free boundary point, i.e. $x_0\in\partial\{u>0\}$ and $u$ is a minimizer of the functional $$J(u):=\int_Ω|\nabla u|^p +λ_+^p\,χ_{\{u>0\}} +λ_-^p\,χ_{\{u\le 0\}}, $$ for some bounded smooth domain $Ω\subset {\mathbb R}^N$ and positive constants $λ_\pm$ with $Λ:=λ_+^p-λ^p_->0$. Here we show the discrete monotonicity of $ϕ_p(r,u,x_0)$ 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.