We prove an Allard-type regularity theorem for free-boundary minimal surfaces in Lipschitz domains locally modeled on convex polyhedra. We show that if such a minimal surface is sufficiently close to an appropriate free-boundary plane, then the surface is 𝐶1,𝛼$C^{1,\alpha }$$C^{1,\alpha }$ graphical over this plane. We apply our theorem to prove partial regularity results for free-boundary minimizing hypersurfaces, and relative isoperimetric regions. © 2022 Wiley Periodicals LLC.

中文翻译：

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局部多面体域中自由边界最小曲面的规律

我们证明了在凸多面体上局部建模的 Lipschitz 域中自由边界最小曲面的 Allard 型正则性定理。我们证明，如果这样的最小表面足够接近适当的自由边界平面，则该表面是𝐶1 、𝛼$C^{1,\alpha }$$C^{1,\alpha }$在这个平面上的图形。我们应用我们的定理来证明自由边界最小化超曲面和相对等周区域的部分正则性结果。© 2022 威利期刊有限责任公司。