SBV functions over a rectifiable current and a compactness theorem
In this paper special functions with bounded variation are introduced in a very general geometric setting, assuming that the domain of the functions is the support \(M\subset{\mathbb{R}}^n\) of an integral current. The paper is the natural continuation of \textit{G. Anzellotti, S. Delladio} and \textit{G. Scianna} [Ann. Mat. Pura Appl., IV. Ser. 170, 257-296 (1996)], where BV functions on \(M\) are defined as those functions whose subgraph \[ E_u:=\{(x,t)\in M\times{\mathbb{R}}:0<t<u(x)\text{ or }u(x)<t<0\} \] (with a suitable orientation induced by the one of \(M\)) has boundary, in the sense of currents, with finite mass. By Federer--Fleming theorem, it is possible in this way to show that BV functions have a \({\mathcal H}^n\)-rectifiable graph \(\Gamma_u\). By looking at the projection \(\pi(\Gamma^*_u)\) on \(M\) of the vertical part \(\Gamma^*_u\) of \(\Gamma_u\), SBV functions can be defined as those BV functions such that \(\pi(\Gamma^*_u)\) is countably \({\mathcal H}^{n-1}\)-rectifiable (this set can also be identified, up to \({\mathcal H}^{n-1}\)-negligible sets, with the approximate discontinuity set of \(u\)). Following basically the strategy of \textit{L. Ambrosio} [Calc. Var. Partial Differ. Equ. 3, No. 1, 127-137 (1995; Zbl 0837.49011)] in this more general setting, a generalized compactness theorem in SBV is proved.
scientific article; zbMATH DE number 1010945
scientific article