Regularity of the free boundary for a semilinear vector-valued minimization problem

In this paper, we consider the following semilinear vector-valued minimization problem $$\min\left\{\int_{D}({|\nabla\mathbf{u}|}^2 + F(|\mathbf{u}|))dx: \ \ \mathbf{u}\in W^{1,2}(D; \mathbb{R}^m) \ \text{and} \ \mathbf{u}=\mathbf{g}\ \text{on} \ \partial D\right\},$$ where $\mathbf{u}: D\to \mathbb{R}^m$ ($m\geq 1$) is a vector-valued function, $D\subset \mathbb{R}^n$ ($n\geq 2$) is a bounded Lipschitz domain, $\mathbf{g}\in W^{1,2}(D; \mathbb{R}^m)$ is a given vector-valued function and $F:[0, \infty)\rightarrow \mathbb{R}$ is a given function. This minimization problem corresponds to the following semilinear elliptic system \begin{equation*} \Delta\mathbf{u}=\frac{1}{2}F'(|\mathbf{u}|)\cdot\frac{\mathbf{u}}{|\mathbf{u}|}\chi_{\{|\mathbf{u}|>0\}}, \end{equation*} where $\chi_A$ denotes the characteristic function of the set A. The linear case that $F'\equiv 2$ was studied in the previous elegant work by Andersson, Shahgholian, Uraltseva and Weiss [Adv. Math 280, 2015], in which an epiperimetric inequality played a crucial role to indicate an energy decay estimate and the uniqueness of blow-up limit. However, this epiperimetric inequality cannot be directly applied to our case due to the more general non-degenerate and non-homogeneous term $F$ which leads to Weiss' boundary adjusted energy does not have scaling properties. Motivated by the linear case, when $F$ satisfies some assumptions, we establish successfully a new epiperimetric inequality, it can deal with term which is not scaling invariant in Weiss' boundary adjusted energy. As an application of this new epiperimetric inequality, we conclude that the free boundary $D\cap \partial\{|\mathbf{u}|>0\}$ is a locally $C^{1,\beta}$ surface near the regular points for some $\beta\in (0,1)$.