Wyniki wyszukiwania

1

On Conditions for Unrectifiability of a Metric Space

100%

Hajłasz P., Malekzadeh S.

Analysis and Geometry in Metric Spaces

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2015

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tom 3

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nr 1

EN

We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

2

Compactness of Special Functions of Bounded Higher Variation

76%

Ambrosio L., Ghiraldin F.

Analysis and Geometry in Metric Spaces

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2013

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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

Some Results on Maps That Factor through a Tree

64%

Züst R.

Analysis and Geometry in Metric Spaces

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2015

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tom 3

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nr 1

EN

We give a necessary and sufficient condition for a map deffned on a simply-connected quasi-convex metric space to factor through a tree. In case the target is the Euclidean plane and the map is Hölder continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over winding number functions. This in particular shows that if the target is the Heisenberg group equipped with the Carnot-Carathéodory metric and the Hölder exponent of the map is bigger than 2/3, the map factors through a tree.

Ograniczanie wyników

3 Analysis and Geometry in Metric Spaces

1 Ambrosio L.

1 Ghiraldin F.

1 Hajłasz P.

1 Malekzadeh S.

1 Züst R.

1 2015

1 2014

1 2013

On Conditions for Unrectifiability of a Metric Space

Compactness of Special Functions of Bounded Higher Variation

Some Results on Maps That Factor through a Tree