We extend Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a sufficient and necessary condition on the integrand to obtain the rectifiability of every \(d\)-dimensional varifold with locally bounded first variation and positive \(d\)-dimensional density. In codimension one, this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane.
Categories: math.AP