Regularity for graphs with bounded anisotropic mean curvature

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