  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 $\beta\in (0,1)$.
