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Given a bounded C² domain in R^n+1 and an integral n-rectifiable varifold V with bounded first variation and bounded generalized mean curvature. Given a C¹ function defined on the boundary of the domain with range (0, ), we assume V has prescribed contact angle with and the tangent cone of V at a point X is a half-hyperplane of density one. Then we can show that the support of V is a C^1, hypersurface with boundary near X for some (0, 1).
Gaoming Wang (Tue,) studied this question.