We study n-dimensional area-minimizing currents T in \(\mathbf {R}^{n+1},\) with boundary \(\partial T\) satisfying two properties: \(\partial T\) is locally a finite sum of \((n-1)\)-dimensional \(C^{1,\alpha }\) orientable submanifolds which only meet tangentially and with same orientation, for some \(\alpha \in (0,1]\); \(\partial T\) has mean curvature \(=h \nu _{T}\) where h is a Lipschitz scalar-valued function and \(\nu _{T}\) is the generalized outward pointing normal of \(\partial T\) with respect to T. Similar to the proof of the Alexandrov reflection principle, we use a geometric maximum principle argument to give a partial boundary regularity result for such currents T. We show that near any point x in the support of \(\partial T,\) either the support of T has very uncontrolled structure, or the support of T near x is the finite union of orientable \(C^{1,\alpha }\) hypersurfaces-with-boundary with disjoint interiors and common boundary points only along the support of \(\partial T\).

我们研究Ñ维面积最小化的电流Ť在\（\ mathbf {R} ^ {N + 1}，\）与边界\（\局部Ť\）满足以下两个属性：\（\局部Ť\）是一个局部\((n-1)\)维\(C^{1,\alpha }\)可定向子流形的有限和，这些子流形仅相切且方向相同，对于某些\(\alpha \in (0,1] \) ; \(\partial T\)具有平均曲率\(=h \nu _{T}\)其中h是 Lipschitz 标量值函数，\(\nu _{T}\)是广义的向外指向的正常\（\局部Ť\）关于T。与 Alexandrov 反射原理的证明类似，我们使用几何最大值原理论证给出了此类电流T的部分边界规则性结果。我们证明在\(\partial T,\)的支持中的任何点x附近要么T的支持具有非常不受控制的结构，要么T在x附近的支持是可定向\(C^{1,\ alpha }\) hypersurfaces-with-boundary 具有不相交的内部和公共边界点仅沿着\(\partial T\) 的支持。