Parametric and non-parametric minima
Let \(u\in BV(G)\) be a local minimum for the non-parametric area functional \(\int \sqrt{1+| Du|^ 2}\). It is well known that the hypograph of u, i.e., \(E=\{x,t)\in G\times {\mathbb{R}}:\quad t<u(x)\}\) is a Caccioppoli set in \(G\times R\) of least perimeter with respect to variations with compact support in \(G\times R\). If \(u\in C^ 2\) this property of E was proved by \textit{W. H. Fleming} [Rend. Circ. Mat. Palermo, II. Ser. 11, 69-90 (1962; Zbl 0107.313)] by applying the divergence theorem to a suitable vector field. Nevertheless, the Fleming argument does not seem to work, since the function is only in BV in this case; in fact, the vector field is not regular enough. The author proves the same result, also extended for more general functionals, by applying a special form of divergence theorem in BV [the author, Traces of bounded vector fields and divergence theorems (to appear)].
scientific article; zbMATH DE number 3899650