Fonseca, Irene; Fusco, Nicola; Marcellini, Paolo Topological degree, Jacobian determinants and relaxation. (English) Zbl 1177.49066 Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 8, No. 1, 187-250 (2005). Summary: A characterization of the total variation \(TV (u,\Omega)\) of the Jacobian determinant \(\det Du\) is obtained for some classes of functions \(u: \Omega \to \mathbb{R}^n\) outside the traditional regularity space \(W^{1,n} (\Omega; \mathbb{R}^{n})\). In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity \(x_{0} \in \Omega\). Relations between \(TV(u, \Omega)\) and the distributional determinant \(\text{Det} Du\) are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps \(u \in W^{1,p} (\Omega; \mathbb{R}^{n}) \cap W^{1,\infty} (\Omega \setminus \{x_{0}\}; \mathbb{R}^{n})\). Cited in 6 Documents MSC: 49Q20 Variational problems in a geometric measure-theoretic setting 35J50 Variational methods for elliptic systems 49J35 Existence of solutions for minimax problems PDFBibTeX XMLCite \textit{I. Fonseca} et al., Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 8, No. 1, 187--250 (2005; Zbl 1177.49066) Full Text: EuDML