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We extend Allard's celebrated rectifiability theorem to the setting ofvarifolds with locally bounded first variation with respect to an anisotropicintegrand. In particular, we identify a sufficient and necessary condition onthe integrand to obtain the rectifiability of every \(d\)-dimensional varifoldwith locally bounded first variation and positive \(d\)-dimensional density. Incodimension one, this condition is shown to be equivalent to the strictconvexity of the integrand with respect to the tangent plane.

Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies