Blow-up of oriented boundaries

Blow-up techniques are used in Analysis and Geometry to investigate local prop-erties of various mathematical objects, by means of their observation at smaller andsmaller scales. In this paper we deal with the blow-up of sets of finite perimeter and,in particular, subsets of Rn (n 2) with prescribed mean curvature in Lp. We provesome general properties of the blow-up and show the existence of a “universal gen-erator”, that is a set of finite perimeter that generates, by blow-up, any other set offinite perimeter in Rn. Then, minimizing sets are considered and for them we derivesome results: more precisely, Theorem 3.5 implies that the blow-up of a set with pre-scribed mean curvature in Ln either gives only area-minimizing cones with the samesurface measure or produces a family of area-minimizing cones whose surface measuresfill a continuum of the real line, while Theorem 3.9 states a sufficient condition for theuniqueness of the tangent cone to a set with prescribed mean curvature in Lp, p > n.