  We prove that $m$-dimensional Lipschitz graphs with anisotropic mean
curvature bounded in $L^p$, $p>m$, are regular almost everywhere in every
dimension and codimension. This provides partial or full answers to multiple
open questions arising in the literature. The anisotropic energy is required to
satisfy a novel ellipticity condition, which holds for instance in a $C^2$
neighborhood of the area functional. This condition is proved to imply the
atomic condition. In particular we provide the first non-trivial class of
examples of anisotropic energies in high codimension satisfying the atomic
condition, addressing an open question in the field. As a byproduct, we deduce
the rectifiability of varifolds (resp. of the mass of varifolds) with locally
bounded anisotropic first variation for a $C^2$ (resp. $C^1$) neighborhood of
the area functional. In addition to these examples, we also provide a class of
anisotropic energies in high codimension, far from the area functional, for
which the rectifiability of the mass of varifolds with locally bounded
anisotropic first variation holds. To conclude, we show that the atomic
condition excludes non-trivial Young measures in the case of anisotropic
stationary graphs.
