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We define a generalized fixed contact angle condition for n-varifold and establish a boundary monotonicity formula. The results are natural generalizations of those for the Neumann boundary condition considered by Grüter-Jost [7].

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We study a singular limit problem of the Allen-Cahn equation with a homogeneous Neumann boundary condition on non-convex domains with smooth boundaries under suitable assumptions for initial data. The main result is the convergence of the time parametrized family of the diffused surface energy to Brakke's mean curvature flow with a generalized right angle condition on the boundary of the domain.

We study a general asymptotic behavior of critical points of a diffused interface energy with a fixed contact angle condition defined on a domain Ω ⊂ R n . We show that the limit varifold derived from the diffused energy satisfies a generalized contact angle condition on the boundary under a set of assumptions.