We establish a partial rectifiability result for the free boundary of a k-varifold V. Namely, we first refine a theorem of Gruter 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^p for some p in [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.
