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Topological degree, Jacobian determinants and relaxation. (English) Zbl 1177.49066
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 \(u \in W^{1,p} (\Omega; \mathbb{R}^{n}) \cap W^{1,\infty} (\Omega \setminus \{x_{0}\}; \mathbb{R}^{n})\).

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
35J50 Variational methods for elliptic systems
49J35 Existence of solutions for minimax problems
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