  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 $\partial \Omega$ and the tangent cone of $V$ at a
point $X \in \partial \Omega$ 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 $\gamma \in (0,1)$.
