On BV functions and essentially bounded divergence-measure fields in metric spaces

Buffa, Vito; Comi, Giovanni E.; Miranda, Marta

doi:10.4171/rmi/1291

Cited by 18 publications

(23 citation statements)

References 38 publications

(109 reference statements)

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“…The following result was proved in [33,Theorem 5.2] by building upon [35,Theorem 6.22]. Theorem 2.13 (Gauss-Green).…”

Section: Let Us Define Snmentioning

confidence: 99%

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Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

Preprint

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This paper studies sharp and rigid isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called N -dimensional RCD(K, N ) spaces (X, d, H N ). Thanks to these results, we determine the asymptotic isoperimetric behaviour for small volumes in great generality, and for large volumes when K = 0 under an additional noncollapsing assumption. Moreover, we obtain new stability results for isoperimetric regions along sequences of spaces with uniform lower Ricci curvature and lower volume bounds, almost regularity theorems formulated in terms of the isoperimetric profile, and enhanced consequences at the level of several functional inequalities.The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences.Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies. Contents 1. Introduction Isoperimetry and lower Ricci curvature bounds Main results Consequences and strategies of the proofs Sharp asymptotic behaviour for small and large volumes Comparison with the previous literature Acknowledgements 2. Preliminaries 2.1. Convergence and stability results 2.2. BV functions and sets of finite perimeter in metric measure spaces 2.3. Sobolev functions, Laplacians and vector fields in metric measure spaces 2.4. Geometric Analysis on RCD spaces 2.5. Localization of the Curvature-Dimension condition 3. The distance function from isoperimetric sets 4. Concavity properties of the isoperimetric profile function and consequences 4.1. Sharp concavity inequalities for the isoperimetric profile 4.2. Fine properties of the isoperimetric profile 4.3. Consequences 5. Stability of isoperimetric sets

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“…The following result was proved in [33,Theorem 5.2] by building upon [35,Theorem 6.22]. Theorem 2.13 (Gauss-Green).…”

Section: Let Us Define Snmentioning

confidence: 99%

“…As shown in [33,Section 5] after [35], for any v ∈ DM ∞ (X) and any set of finite perimeter E, there exist measures…”

Section: Let Us Define Snmentioning

confidence: 99%

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Antonelli¹,

Pasqualetto²,

Pozzetta³

et al. 2022

Preprint

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show abstract

“…The convergence of (6.22) to H N −1 {d ∂X = s} is a classical statement, see for instance [ADMG17,BCM21] for the weak convergence to the perimeter of {d ∂X ≤ s} and [ABS19,BPS19] for the identification between perimeter and (N − 1)-dimensional Hausdorff measure.…”

Section: The First Ingredient Says That Any Ballmentioning

confidence: 99%

The metric measure boundary of spaces with Ricci curvature bounded below

Bruè¹,

Mondino²,

Semola³

2022

Preprint

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“…For the following definitions and properties, we are going to follow [12], see also [7,8]. While in the literature derivations are explained for Lipschitz functions with bounded support, we write Lip c instead since the underlying space X in this paper is proper.…”

Section: The Space Bv Via Derivationsmentioning

confidence: 99%

“…To overcome this obstacle and to be able to give a suitable definition of a parabolic function space, we make use of an alternative approach: in the Euclidean case, a function u : (0, T ) → BV( ) is said to be weakly measurable if the mapping t → u(t)div(φ) dx is measurable with respect to the Lebesgue measure on (0, T ) for every test vector field φ. On a metric measure space, one can explain a divergence operator for derivations d and find an equivalent characterization of BV which relies on an integration by parts formula that is based on derivations and their divergence, see [7,8,12] and Sect. 2 in this paper.…”

Section: Introductionmentioning

confidence: 99%

Existence of parabolic minimizers to the total variation flow on metric measure spaces

Buffa¹,

Collins

Camacho

2022

manuscripta math.

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We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$ u 0 on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$ Ω × ( 0 , T ) with $$\Omega \subset {\mathcal {X}}$$ Ω ⊂ X an open set and $$T > 0$$ T > 0 , we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$ L w 1 ( 0 , T ; BV ( Ω ) ) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$ BV -valued parabolic function spaces. We argue completely on a variational level.

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On BV functions and essentially bounded divergence-measure fields in metric spaces

Cited by 18 publications

References 38 publications

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

The metric measure boundary of spaces with Ricci curvature bounded below

Existence of parabolic minimizers to the total variation flow on metric measure spaces

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