Rectifiability of the free boundary for varifolds

Abstract. We establish a partial rectifiability result for the free boundary of a k-varifold V . Namely, we first refine a theorem of Grüter and Jost by showing that the first variation of a general varifold with free boundary is a Radon measure. Next we show that if the mean curvature H of V is in L for some p ∈ [1, k], then the set of points where the k-density of V does not exist or is infinite has Hausdorff dimension at most k − p. We use this result to prove, under suitable assumptions, that the part of the first variation of V with positive and finite (k− 1)-density is (k− 1)-rectifiable.