We present our recent extension of 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.

We can apply this result to the minimization of anisotropic energies among families of d-rectifiable closed subsets of $\mathbb{R}^n$. Easy corollaries of this compactness result are the solutions to three formulations of the Plateau problem: one introduced by Reifenberg, one proposed by Harrison and Pugh and another one studied by Guy David.

Moreover, we apply the rectifiability theorem to prove an anisotropic counterpart of Allard's compactness result for integral varifolds.