Compactness of Special Functions of Bounded Higher Variation

Given an open set Omega subset of R-m and n > 1, we introduce the new spaces GB(n) V( Omega) of Generalized functions of bounded higher variation and GSB(n) V(Omega) 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 J u need not have finite mass: roughly speaking, finiteness of mass is not required for the (m - n)-dimensional part of J(u), but only finiteness of size. In the space GSB(n)V 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].