Blow-up of oriented boundaries
The author investigates blow-ups of sets of finite perimeter in \({\mathbb R}^{n}\) (\(n\geq 2\)). After some preliminary definitions and observations, the author proves the interesting fact that there exists a set \(M\) of finite perimeter such that the family of all possible blow-ups of \(M\) at the origin is equal to the family of all sets of finite perimeter. The author then considers weakly-minimizing sets and shows that the following alternative holds: Either (1) the density at the origin is defined and every blow-up at the origin is an area-minimizing cone, or (2) the lower density at the origin, \(\ell\), is strictly less than the upper density at the origin, \(L\), and, for each \(\lambda\in[\ell,L]\), there is a blow-up that is an area-minimizing cone with density \(\lambda\). The final result of the paper provides a sufficient condition for the uniqueness of the blow-up. This sufficient condition is described in terms of \[ \alpha_{E}(r) = r^{1-n} P(E,B_{r}(0)) \] and \[ \psi(E,r) = P(E,B_{r}(0)) - \inf\{P(F,B_{r}(0)): F\triangle E\subset\subset B_r(0)\}. \] Specifically, the sufficient condition is that there exist positive \(R\), \(c\), \(\varepsilon\), and \(\beta\) such that \(\alpha_{E}(r)\) is a \(C^{0,\beta}\) function on \((0,R)\) and that \[ \psi(E,r)\leq c r^{n-1+\varepsilon} \] holds on \((0,R)\).
scientific article; zbMATH DE number 1544945
scientific article