Topological degree, Jacobian determinants and relaxation

Fonseca, Irene ; Fusco, Nicola ; Marcellini, Paolo

Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 1, pp. 187-250

Résumé

A characterization of the total variation $TV(u, \Omega)$ of the Jacobian determinant $\det Du$ is obtained for some classes of functions $u : \Omega \rightarrow \mathbb{R}^{n}$ 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 $u\in W^{1, p}(\Omega; \mathbb{R}^{n})\cap W^{1, \infty}(\Omega\backslash \{x_0\}; \mathbb{R}^{n})$.