Functions of bounded variation on “good” metric spaces

By Michele Miranda Jr.. 2003

341 citations recorded by the source corpus.

Abstract In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincare inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L 1 topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of Γ -convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of BV functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot–Caratheodory spaces.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f2e7703ee76cb32dcf9b18d7b4e2b890e5f623c3

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In Geometry and Topology

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Citations in the corpus

Pointwise properties of functions of bounded variation in metric spaces (2014, neutral): “Next we recall the definition and basic properties of functions of bounded variation on metric spaces, see [14].”
The globalization theorem for the Curvature-Dimension condition (2016, background): “10 (1)), possibly using a cut-off Lipschitz function over Br(o), we may assume that φs has bounded total variation measure ‖Dφs‖ (we refer to [53] and [9] for all missing notions and background regarding BV-functions on metric-measure spaces).”
Equivalent definitions of BV space and of total variation on metric measure spaces (2014, methodology): “Recall that, according to the notion of BV function given in [27], a function f ∈ L(1)(X, d,m) belongs to BV∗(X, d,m) if there exist locally Lipschitz functions fn convergent to f in L (1)(X) such that”
Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces. (2019, background): “In [3] (see also [41] for the case of locally compact spaces) it is proven that this set function is the restriction to open sets of a finite Borel measure that we call total variation of f and still denote by |Df |.”
Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, background): “It is proved in [55] that, for any f 2 BV.”
The BV-capacity in metric spaces (2010, background): “Notice that in [24] the functions of bounded variation are defined in terms of the lower pointwise dilation.”
Geometric Implications of the Poincaré Inequality (2007, background): “spheres. If the measure is Ahlfors Q‐regular, for some Q > 1, then the results of [1] and [ 15 ] yield upper”
Minimizers of the variable exponent, non-uniformly convex Dirichlet energy (2008, neutral): “As a remark, we note some recent efforts to extend BV spaces to the framework of metric measure spaces, [8,9,45].”
Lebesgue points and capacities via the boxing inequality in metric spaces (2008, background): “Note that if u is a locally Lipschitz continuous function, its local Lipschitz constant is an upper gradient of u. We observe that all results of [ Mi ] hold for upper gradients as well.”
Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. (2011, methodology): “Following the approach in [19], for any open set A ⊂ X we can define |Df |(A) := inf {”