In this paper we address a one phase minimization problem for a functional that includes the perimeter of the positivity set. It also includes three terms, the first one is $\int fu$ and the second ${\int }_{u>0}h$ where $f$ and $h$ are bounded functions. The third term is $\int G\left(|\nabla u|\right)$ where $G$ is a smooth convex function. This term generalizes the integral of the ${|\nabla u|}^p$. As a consequence of our results we find that, when $f\le 0$, there exists a nonnegative minimizer. Moreover, every nonnegative minimizer is Lipschitz continuous, it is a solution to ${\Delta }_{G}u=f$ in $\{u>0\}$ and satisfies that $H=\Phi \left(|\nabla u|\right)-h$ on the reduced free boundary, ${\partial }_{red}\{u>0\}$ which, as a consequence, is proved to be as smooth as the data allow. Here $\Phi \left(t\right)=tg\left(t\right)-G\left(t\right)$ ($g=G^{\prime }$) and $H$ is the mean curvature of the free boundary.

在本文中，我们解决了一个包含正集周长的泛函的单相最小化问题。它还包括三个术语，第一个是$\int F你$ 第二个 ${\int }_{你>0}H$ 在哪里 $F$ 和 $H$是有界函数。第三项是$\int G\left(|\nabla 你|\right)$ 在哪里 $G$是一个光滑的凸函数。该术语概括了${|\nabla 你|}^{磷}$. 根据我们的结果，我们发现，当$F\le 0$，存在一个非负极小值。此外，每个非负极小值都是 Lipschitz 连续的，它是一个解${\Delta }_G你=F$ 在 $\{你>0\}$ 并满足 $H=\Phi \left(|\nabla 你|\right)-H$ 在缩减的自由边界上， ${\partial }_{r\mathrm{电子}d}\{你>0\}$因此，它被证明是数据所允许的那样平滑。这里$\Phi \left(吨\right)=吨G\left(吨\right)-G\left(吨\right)$ ($G=G^{\prime }$） 和 $H$ 是自由边界的平均曲率。