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

Abstract

In this paper, we consider the following vector-valued minimization problem$\min \left\{\underset{D}{\int }\right(|\nabla u{|}^2+F(\left|u\right|\left)\right)dx:u\in W^{1,2}(D;R^m)\text{and}u=g\text{on}\partial D\}$ where $u:D\to R^m$ ($m\ge 1$) is a vector-valued function, $D\subset R^n$ ($n\ge 2$) is a bounded Lipschitz domain, $g\in W^{1,2}(D;R^m)$ is a given vector-valued function and $F:[0,\infty )\to R$ is a given function. This minimization problem corresponds to the following semilinear elliptic system$\Delta u=\frac{1}{2}{F}^{\prime }\left(\right|u\left|\right)\frac{u}{\left|u\right|}{\chi }_{\left\{\right|u|>0\}},$ where ${\chi }_A$ denotes the characteristic function of the set A. The linear case that $F^{\prime }\equiv 2$ was studied in the previous elegant work by Andersson et al. (2015) [3], 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's energy functional 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's energy functional. As an application of this new epiperimetric inequality, we conclude that the free boundary $D\cap \partial \left\{\right|u|>0\}$ is a locally $C^{1,\beta }$ surface near the regular points for some $\beta \in (0,1)$.