Compactness of Special Functions of Bounded Higher Variation

Luigi Ambrosio; Francesco Ghiraldin

Analysis and Geometry in Metric Spaces (2013)

  • Volume: 1, page 1-30
  • ISSN: 2299-3274

Abstract

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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].

How to cite

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Luigi Ambrosio, and Francesco Ghiraldin. "Compactness of Special Functions of Bounded Higher Variation." Analysis and Geometry in Metric Spaces 1 (2013): 1-30. <http://eudml.org/doc/266762>.

@article{LuigiAmbrosio2013,
abstract = {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].},
author = {Luigi Ambrosio, Francesco Ghiraldin},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Higher codimension singularities; nonlinear elasticity; geometric measure theory; distributional jacobian; flat currents; special bounded variation; compactness; bounded higher variation; Mumford-Shah; free discountinuity; higher codimension singularities; distributional Jacobian; Mumford-Shah type funtional},
language = {eng},
pages = {1-30},
title = {Compactness of Special Functions of Bounded Higher Variation},
url = {http://eudml.org/doc/266762},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Luigi Ambrosio
AU - Francesco Ghiraldin
TI - Compactness of Special Functions of Bounded Higher Variation
JO - Analysis and Geometry in Metric Spaces
PY - 2013
VL - 1
SP - 1
EP - 30
AB - 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].
LA - eng
KW - Higher codimension singularities; nonlinear elasticity; geometric measure theory; distributional jacobian; flat currents; special bounded variation; compactness; bounded higher variation; Mumford-Shah; free discountinuity; higher codimension singularities; distributional Jacobian; Mumford-Shah type funtional
UR - http://eudml.org/doc/266762
ER -

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