Mean convexity of the zero set of symmetric minimal surfaces

Abstract

Let $\Omega \subset \mathbb {R}^{n}$ be a bounded open set, $\alpha > 0$ a given constant, and u a bounded local minimizer of the functional

$$\begin{aligned} \mathcal {F}(u) := \int _{\Omega } u^{\alpha } \sqrt{1 + |Du|^{2}} \, dx \end{aligned}$$

in the class $BV_{+}^{1 + \alpha } (\Omega ) := \{ u \in L^{1+ \alpha } (\Omega ) : u \ge 0,\ u^{1 + \alpha } \in BV(\Omega ) \}$. We show that minimizers are elements of $W^{1,1}_{loc}(\Omega )$ and that the coincidence set $\{u = 0\}$ is a set of locally finite perimeter in $\Omega$, which, in case $\alpha \ge 1$, has nonnegative inward mean curvature in the variational sense, i.e. is mean convex. In particular, if $\alpha \ge 1$, we prove the inequality

$$\begin{aligned} \int _{\Omega } |D\chi _{\{u = 0 \} \cap E}| \le \int _{\Omega } |D\chi _{E}| - \int _{E \cap \{u > 0\}} \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \, dx \end{aligned}$$

for all sets $E \subset \subset \Omega$ of finite perimeter.