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arXiv · 1210.6473

Compactness of special functions of bounded higher variation

Luigi Ambrosio·Francesco Ghiraldin
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Abstract

Given an open set \Omega\subset\R^m and n>1, we introduce the new spaces GB_nV(\Omega) of Generalized functions of bounded higher variation and GSB_nV(\Omega) of Generalized special functions of bounded higher variation that generalize, respectively, the space B_nV introduced by Jerrard and Soner and the corresponding SB_nV space studied by De Lellis. In this class of spaces, which allow 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 GSB_nV 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.

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Luigi Ambrosio, Francesco Ghiraldin. 2012-10-24. Compactness of special functions of bounded higher variation. https://doi.org/10.2478/agms-2012-0001

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