Allard-Type Regularity for Varifolds with Prescribed Contact Angle

Given a bounded $C^2$ domain in ${\mathbb{R}}^{n+1}$ and an integral $n$-rectifiable varifold $V$ with bounded first variation and bounded generalized mean curvature. Given a $C^1$ function $\theta$ defined on the boundary of the domain with range $(0,\pi )$, we assume $V$ has prescribed contact angle $\theta$ with $\mathrm{\partial }\mathrm{\Omega }$ and the tangent cone of $V$ at a point $X\in \mathrm{\partial }\mathrm{\Omega }$ is a half-hyperplane of density one. Then we can show that the support of $V$ is a $C^{1,\gamma }$ hypersurface with boundary near $X$ for some $\gamma \in (0,1)$.