In this talk, we discuss a boundary regularity result of \emph{Allard type} for varifolds meeting a domain boundary at a prescribed contact angle.
Let $\Omega \subset \mathbb{R}^{n+1}$ be a bounded $C^2$ domain, and let $V$ be an integral $n$-rectifiable varifold in $\Omega$ with bounded first variation and generalized mean curvature in $L^\infty$.
Suppose that a $C^1$ contact angle function $\theta : \partial \Omega \to (0,\pi)$ is prescribed, and that $V$ satisfies this boundary angle condition along $\partial \Omega$.
Assuming further that the tangent cone of $V$ at a boundary point $X \in \partial \Omega$ is a half-hyperplane of density one, we establish a boundary regularity theorem:
in a neighborhood of $X$, the support of $V$ is a $C^{1,\gamma}$ hypersurface with boundary for some $\gamma \in (0,1)$.
This extends the classical interior regularity theorem of Allard to situations with nontrivial boundary contact conditions and provides a geometric framework for studying variational problems with prescribed contact angles.