Anisotropic counterpart of Allard’s rectifiability theorem and Plateau problem.

We present our extension of Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded anisotropic first variation. We identify a necessary and sufficient condition on the integrand for its
validity and we discuss the connections of this condition to Almgren's ellipticity. We apply this result to the set-theoretic anisotropic Plateau problem, obtaining solutions to three different formulations: one introduced by
Reifenberg, one proposed by Harrison and Pugh and another one studied by David. Moreover, we apply the rectifiability theorem to prove an anisotropic counterpart of Allard's compactness result for integral varifolds.