  We consider the minimization problem of an anisotropic energy in classes of
$d$-rectifiable varifolds in $\mathbb R^n$, closed under Lipschitz deformations
and encoding a suitable notion of boundary. We prove that any minimizing
sequence with density uniformly bounded from below converges (up to
subsequences) to a $d$-rectifiable varifold. Moreover, the limiting varifold is
integral, provided the minimizing sequence is made of integral varifolds with
uniformly locally bounded anisotropic first variation.
