Sobolev and bounded variation functions on metric measure spaces

Abstract

− ∇f, ∆∇f = Hessf 2 + Ric(∇f, ∇f). We will refer to this theory as "Eulerian theory", in the sense that it involves gradient, Laplacian etc., and all these concepts make sense also in the abstract setup provided by Dirichlet forms and the associated heat semigroup (whose fundamental generator is precisely the Laplacian). One of the main advantages of Bakry-Émery approach is that it is wellsuited to get useful functional inequalities even in sharp form (e.g., Li-Yau inequality). The other side of the theory is concerned with optimal transport and was developed independently by Lott-Villani and Sturm. We will refer to their approach as "Lagrangian

Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces

“…The latter inequality answers a question raised in [24] and later in [3]. (X) and even in Lip(X), since g is bounded.…”

“…Next we recall the de nition and basic properties of functions of bounded variation on metric spaces, see [1], [3] and [24]. For u ∈ L loc (X), we de ne the total variation of u as…”

“…See [3] and [22] for some examples of local spaces, and [23, Example 5.2] for an example of a space that is not local, despite having a doubling measure and a Poincaré inequality. The assumption E ⊂ E can, in fact, be removed as follows.…”

“…where [1,Theorem 5.3] and [3,Theorem 4.6]. The constant c P is related to the Poincaré inequality, see below.…”

“…For f (t) = t, this gives the de nition of functions of bounded variation, or BV functions, on metric measure spaces, see [1], [3] and [24]. For f (t) = √ + t , we get the generalized surface area functional, which has been considered previously in [17] and [18].…”