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Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies
Guido De Philippis, Antonio De Rosa and Francesco Ghiraldin
In this paper 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 necessary and sufficient 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.