  In this paper we investigate the "area blow-up" set of a sequence of smooth
co-dimension one manifolds whose first variation with respect to an anisotropic
integral is bounded. Following the ideas introduced by White in (J.
Differential Geom., 2016), we show that this set has bounded (anisotropic) mean
curvature in the viscosity sense. In particular, this allows to show that the
set is empty in a variety of situations. As a consequence, we show boundary
curvature estimates for two dimensional stable anisotropic minimal surfaces,
extending the results of (Invent. Math., 1987).
