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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

CiteScore 2017: 0.65

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Compactness of Special Functions of Bounded Higher Variation

Luigi Ambrosio / Francesco Ghiraldin
Published Online: 2013-01-04 | DOI: https://doi.org/10.2478/agms-2012-0001


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].

Keywords: Higher codimension singularities; nonlinear elasticity; geometric measure theory; distributional jacobian; flat currents; special bounded variation; compactness; bounded higher variation; Mumford-Shah; free discountinuity

MSC: 49Q20; 49J45; 49Q15

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About the article

Received: 2012-10-16

Accepted: 2012-12-07

Published Online: 2013-01-04

Citation Information: Analysis and Geometry in Metric Spaces, Volume 1, Pages 1–30, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2012-0001.

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