Finite perimeter sets with reduced boundary almost equal to their closure that are volume-constrained critical points of a C^3 elliptic anisotropic perimeter must be finite unions of disjoint equal-radius Wulff shapes.
The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals
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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).
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Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets
Finite perimeter sets with reduced boundary almost equal to their closure that are volume-constrained critical points of a C^3 elliptic anisotropic perimeter must be finite unions of disjoint equal-radius Wulff shapes.