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 $\gamma \in (0,1)$
This paper establishes a boundary Allard theorem for varifolds with prescribed contact angle, including variable angles and ambient metrics of low regularity.
Boundary regularity
If a boundary point admits a multiplicity-one tangent half-hyperplane, then the support is locally a $C^{1,\gamma}$ hypersurface with boundary for some $\gamma\in(0,1)$.
Contact angle
On the regular support, the geometric contact angle is the prescribed angle (apart from the explicitly identified orthogonal alternative); the ambiguity disappears under a natural nonorthogonality condition.
Density criterion
For two-dimensional varifolds, or in every dimension when the angle is sufficiently close to $90^\circ$, a quantitative boundary density bound implies the required regularity.
The proof is carried out in a flattened half-space equipped with a $C^1$ metric, so it remains intrinsic to the codimension-one contact-angle problem. It combines a boundary monotonicity formula with an $L^2$ height and tilt estimate, a blow-up analysis, and geometric excess decay. A reflection idea inspired by Leon Simon explains how the boundary term and the capillary angle are incorporated into a stationary comparison object.
My research interests include Geometric Analysis and Partial Differential Equations.