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On the Regularity of Alexandrov Surfaces with Curvature Bounded Below

100%
Ambrosio L., Bertrand J.
Analysis and Geometry in Metric Spaces
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2016
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tom 4
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nr 1
EN
In this note, we prove that on a surface with Alexandrov’s curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [1, 2), locally belong to W1,p out of a discrete singular set. This result is based on Reshetnyak’s work on the more general class of surfaces with bounded integral curvature.
2
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Compactness of Special Functions of Bounded Higher Variation

100%
Ambrosio L., Ghiraldin F.
Analysis and Geometry in Metric Spaces
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2013
|
tom 1
1-30
EN
Given an open set Ω ⊂ Rm and n > 1, we introduce the new spaces GBnV(Ω) of Generalized functions of bounded higher variation and GSBnV(Ω) of Generalized special functions of bounded higher variation that generalize, respectively, the space BnV introduced by Jerrard and Soner in [43] and the corresponding SBnV space studied by De Lellis in [24]. In this class of spaces, which allow as in [43] the description of singularities of codimension n, the distributional jacobian Ju need not have finite mass: roughly speaking, finiteness of mass is not required for the (m−n)-dimensional part of Ju, but only finiteness of size. In the space GSBnV we are able to provide compactness of sublevel sets and lower semicontinuity of Mumford-Shah type functionals, in the same spirit of the codimension 1 theory [5,6].
3
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Some Fine Properties of BV Functions on Wiener Spaces

81%
Ambrosio L., Miranda Jr. M., Pallara D.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
|
nr 1
EN
In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.
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