Some sharp Sobolev inequalities on $ BV({\mathbb{R}}^n) $

Abstract

In this paper, some sharp Sobolev inequalities on $ BV({\mathbb{R}}^n) $, the space of functions of bounded variation on $ {\mathbb{R}}^n $, $ n\geq 2 $, are deduced through the $ L_p $ Brunn-Minkowski theory. We will prove that these inequalities can all imply the sharp Sobolev inequality on $ BV({\mathbb{R}}^n) $.

Keywords

• sobolev inequalities
• functions of bounded variation
• optimal constants